The Mean Value Theorem

12 Key excerpts on "The Mean Value Theorem"

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  This leads to The Mean Value Theorem, which says: The Mean Value Theorem: suppose that f is continuous on [ a, b ] and differentiable on ( a, b ). Then there’s at least one number c in ( a, b ) such that f 0 ( c ) = f ( b ) -f ( a ) b -a . It seems a little weird, but it actually makes sense. You see, if f ( t ) is your position at time t , and you start and finish at times a and b , respectively, then what is your average velocity? The displacement is f ( b ) -f ( a ), while the time taken is b -a , so the quantity on the right-hand side of the above equation is just your average velocity. On the other hand, f 0 ( c ) is your instantaneous velocity at time c . The Mean Value Theorem says that there is at least one time c where your instantaneous velocity equals your average velocity over the whole journey. Let’s look at a picture of the situation. Suppose your function f looks like this: * Again, all this assumes—very reasonably—that your car’s motion is differentiable! 234 • The Derivative and Graphs a b c 0 c 1 ( a, f ( a )) ( b, f ( b )) The dashed line joining ( a, f ( a )) and ( b, f ( b )) has slope f ( b ) -f ( a ) b -a . According to The Mean Value Theorem, there is some tangent whose slope equals this quantity; that is, some tangent is parallel to the dashed line. In the above picture, there are actually two tangents that work—the x -coordinates are at c 0 and c 1 . Either one would be an acceptable candidate for the number c in the theorem. The Mean Value Theorem looks a lot like Rolle’s Theorem. In fact, the conditions for applying the two theorems are almost the same. In both cases, your function f has to be continuous on a closed interval [ a, b ] and differen-tiable on ( a, b ). Rolle’s Theorem also requires that f ( a ) = f ( b ), but The Mean Value Theorem doesn’t require that.

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  Understanding Real Analysis

  • Paul Zorn(Author)
  • 2017(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  4.3 The Mean Value Theorem

  The Mean Value Theorem (MVT) is a marquee result of real analysis. As the intermediate value theorem (IVT) and the extreme value theorem (EVT) do for continuous functions, The Mean Value Theorem addresses a natural and fundamental property of differentiable functions.

  Figure 4.5 The Mean Value Theorem illustrated.

  Theorem 4.9 (The Mean Value Theorem). Suppose

  f : [ a , b ] → R

  is continuous on

  [ a , b ]

  and differentiable on

  ( a , b )

  . Then there exists

  c ∈ ( a , b )

  such that

  f ′

  ( c )

  =

  f ( b ) - f ( a )

  b - a

  .

  Like the IVT and the EVT, the MVT says that a certain function–f ′ , this time—assumes a certain “mean” (or average”) value. Unexciting as this might seem, it turns out to be a big deal, with surprisingly broad and deep implications. We explore several before proving the theorem.

  The graphical view. The MVT equation says something about slopes on the graph of f :At some point c between a and b , the tangent line is parallel to the secant line joining

  ( a , f ( a ) )

  and

  ( b , f ( b ) )

  . Figure 4.5 illustrates this. The (linear) secant line function, labeled L (x ) in the picture, will play a role in the proof.

  The horizontal case: Rolle’s theorem. If we add to the other hypotheses the requirement that f (a ) = f (b ), then the conclusion takes a slightly simpler form:

  Theorem 4.10 (Rolle’s theorem). Suppose

  f : [ a , b ] → R

  is continuous on

  [ a , b ]

  and differentiable on

  ( a , b )

  , and that f (a ) = f (b ) . Then f ’ (c ) = 0 for some

  c ∈ ( a , b )

  .

  In graphical terms, the new hypothesis says that the graph has the same height at x  = a and at x  = b ; the conclusion says that the graph has a horizontal tangent somewhere in between. Figure 4.6 shows two such horizontal tangents. More than one possible c is fine with Rolle.

  Figure 4.6 Rolle’s theorem: two possible c ‐values.

  Car talk. If f (t ) is the position of a car at time t

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  Introduction to Differential Calculus

  Systematic Studies with Engineering Applications for Beginners

  • Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  4

  The conclusions of the MVT are intuitively appealing and its hypotheses are naturally expected. Further, its applications will be found marvelously tangible (i.e., fitting with experience). The student will find in the MVT, an ever present tool just waiting to be applied, both in proving theorems and in solving problems.

  The adjective “mean ” carries both the notions “between ” and “average ”, each of which gives a significant clue to the basic idea in the theorem. What the MVT does, is single out a derivative value that plays the role of an average derivative value, and this derivative value is attained at a point strictly between the end points of the interval domain of the function.

  Consider a continuous function ,

  which is differentiable at every point of the open interval (a , b ).5

  Figure 20.12

  What the MVT does is to identify the difference quotient with the derivative f ′(x ) evaluated at a point (say) c , lying strictly between a and b .

  That is,

  It must be clearly understood once and for all that the location of c is not really pinpointed ; we only know that it lies somewhere inside an open interval (a , b ). But, the interesting fact is the mere knowledge that c is a mean point (i.e., it lies strictly between the end points of the interval) shows the real power behind the theorem and its applications . Of course, the exact location of c can be found in some cases (as we have seen in some solved examples) but in general, it is never needed in any application.

  Many important concepts in mathematics are based on Existence Theorems

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  Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  n EXAMPLE 4 If an object moves in a straight line with position function s - f s t d, then the average velocity between t - a and t - b is f sbd 2 f sad b 2 a and the velocity at t - c is f 9 scd. Thus The Mean Value Theorem (in the form of Equation 1) tells us that at some time t - c between a and b the instantaneous velocity f 9 scd is equal to that average velocity. For instance, if a car traveled 180 km in 2 hours, then the speedometer must have read 90 kmyh at least once. In general, The Mean Value Theorem can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval. n The main significance of The Mean Value Theorem is that it enables us to obtain infor- mation about a function from information about its derivative. The next example pro- vides an instance of this principle. Lagrange and The Mean Value Theorem The Mean Value Theorem was first for- mulated by Joseph-Louis Lagrange (1736–1813), born in Italy of a French father and an Italian mother. He was a child prodigy and became a professor in Turin at the age of 19. Lagrange made great contributions to number theory, theory of functions, theory of equations, and analytical and celestial mechanics. In particular, he applied cal- culus to the analysis of the stability of the solar system. At the invitation of Frederick the Great, he succeeded Euler at the Berlin Academy and, when Frederick died, Lagrange accepted King Louis XVI’s invitation to Paris, where he was given apartments in the Louvre and became a professor at the Ecole Polytechnique. Despite all the trap- pings of luxury and fame, he was a kind and quiet man, living only for science. FIGURE 6 y=˛-x B x y c 2 O Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

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  Real Analysis

  With Proof Strategies

  • Daniel W. Cunningham(Author)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Theorem 5.2.1 asserts that f ′ (c) = 0. As portrayed in Figure 5.2, Rolle’s Theorem 5.2.2 implies that the graph of any differentiable function which has the same value at two distinct points must have a tangent line that is parallel to the x -axis. Using Theorem 5.2.2, we will now state and prove The Mean Value Theorem, Figure 5.2 Illustration for Theorem 5.2.2. Theorem 5.2.3 (Mean Value Theorem). Let f : [ a, b ] → ℝ be a continuous function that is differentiable on the open interval (a, b). Then there is a point c ∈ (a, b) such that f ′ (c) = f (b) − f (a) b − a. Proof. Let f : [ a, b ] → ℝ be a continuous function. Suppose that f is differentiable on the open interval (a, b). Define the function h : [ a, b ] → ℝ by h (x) = f (x) − (f (b) − f (a) b − a (x − a) + f (a)) ​. (5.8) Note that h is a continuous on [ a, b ] and differentiable on (a, b). Moreover, h (a) = 0 and h (b) = 0. Theorem 5.2.2 implies that there is a c ∈ (a, b) such that h ′ (c) = 0. Equation (5.8) implies that h ′ (c) = f ′ (c) − f (b) − f (a) b − a = 0. Thus, f ′ (c) = f (b) − f (a) b − a. Figure 5.3 offers a geometric interpretation of Theorem 5.2.3. The value f (b) − f (a) b − a is the slope of the secant line between the points (a, f (a)) and (b, f (b)). The theorem states that there is a c ∈ (a, b) such that the slope of the associated tangent line, f ′ (c), is equal to the slope of this secant line. Figure 5.3 Illustration for Theorem 5.2.3. The Mean Value Theorem can be used to derive inequalities. For example, let 0 ≤ a < b. We will prove that b 3 − a 3 < 3 b 2 (b − a). Let f : [ a, b ] → ℝ be defined by f (x) = x 3. Then f is continuous on [ a, b ] and differentiable on (a, b). By Theorem 5.2.3, there is a c ∈ (a, b) such that b 3 − a 3 = 3 c 2 (b − a)

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  Essential Calculus with Applications

  • Richard A. Silverman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  m .

  3.43. The Mean Value Theorem

  a. If f (a ) ≠ f (b ), we can no longer assert that the curve (8) has a horizontal tangent at some intermediate point. However, we can now assert that at some intermediate point the curve has a tangent with the same slope as the chord joining its end points A = (a , f (a )) and B = (b , f (b )), that is, a tangent with slope

  as illustrated by Figure 5 .

  Figure 4 .

  Figure 5 .

  THEOREM (Mean value theorem). Let f be continuous in the closed interval [a , b ] and differentiable, with derivative f ′, in the open interval (a , b ). Then there is a point c (a , b ) such that

  or equivalently

  Proof . We introduce a new function

  g (x ) = f (x ) − kx ,

  choosing the constant k in such a way that g (x ) has the same value at both end points a and b . Then k must satisfy the equation

  g (a ) = f (a ) − ka = f (b ) − kb = g (b ),

  with solution

  With this choice of k , g (x ) satisfies all the conditions of Rolle’s theorem. But then there is a point c (a , b ) such that

  g ′(c ) = f ′(c ) − k = 0,

  that is, such that

  b. COROLLARY (Mean value theorem in increment form) . If f is differentiable in an interval I containing the points x and x + Δx , then the increment of f at x can be written in the form

  where 0 < α < 1.

  Proof . Here it is not necessary to state explicitly that f is continuous in I , since this follows automatically from the assumption that f is differentiable in I (Sec. 2.66 ). Choosing a = x , b = x + Δx in (10) , we obtain

  where c lies between x and x + Δx , regardless of the sign of Δx

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  The Fundamentals of Mathematical Analysis

  • G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 6 BASIC THEOREMS OF DIFFERENTIAL CALCULUS § 1. Mean value theorems ___ 100. Fermat's theorem. The knowledge of the derivative (or a number of derivatives) of a function makes it possible to draw conclusions regarding the function itself. The basis of various appli-cations of the concept of a derivative (see Chapters 7 and 13) rests on certain simple but important theorems and formulae to which this chapter is devoted. We begin by examining a statement which is linked with the name of Fermât 1 . Of course, he did not announce it in the form in which we present it here (Fermât did not know of the concept of a de-rivative); however, our form re-establishes the essence of Fermafs device as applied by him to determining the greatest and the smallest values of a function (see Chapter 14). FERMAT'S THEOREM. Suppose that a function f(x) is defined in an interval 9C and that it takes at an interior point of the interval its greatest (smallest) value. If at this point there exists the finite derivative f '(c), then, necessarily, f (c) = 0. Proof For definiteness let f(x) take at a point c its greatest value; hence for all x from 9C we have f(*)c the expression X — C and hence passing to the limit, x-»c + 0, we obtain /'(c)<0. (1) If now x < c, then x—c passing to the limit, x->c — 0, we have fc) = 0. (2) Comparing relations (1) and (2) we arrive at the required result /'(c) = 0. Remark. The reasoning carried out above proves, essentially, that at the considered point c an infinite (two-sided) derivative cannot exist.

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  Introduction to Functional Equations

  • Prasanna K. Sahoo, Palaniappan Kannappan(Authors)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 16 Mean Value Type Functional Equations 16.1 Introduction In this chapter, we will examine some functional equations that arise from the Lagrange mean value theorem. We call these functional equa-tions the mean value type of functional equations. The study of this type of functional equations was started by Acz´ el (1985), Haruki (1979) and Kuczma (1991a, 1991b). A detailed account on mean value type func-tional equations can be found in the book Mean Value Theorems and Functional Equations by Sahoo and Riedel (1998). 16.2 The Mean Value Theorem The Mean Value Theorem (MVT) whose proof can be in the book Mean Value Theorems and Functional Equations by Sahoo and Riedel (1998) (see pages 25–27) is the following. Theorem 16.1. Suppose the real-valued function f is continuous on the closed interval [ a, b ] and is differentiable on the open interval ( a, b ) . Then there exists η ∈ ( a, b ) such that f 0 ( η ) = f ( b ) -f ( a ) b -a . (16.1) The equation of the tangent line at the point ( η, f ( η )) is given by y = ( x -η ) f 0 ( η ) + f ( η ) . (16.2) The equation of the secant line joining the points ( a, f ( a )) and ( b, f ( b )) 243 244 Introduction to Functional Equations is given by y = ( x -a ) f ( b ) -f ( a ) b -a + f ( a ) . (16.3) If the secant line is parallel to the tangent line at η (see Figure 16.1), then f 0 ( η ) = f ( b ) -f ( a ) b -a . (16.4) -6 y 0 x tangent line secant line a η b . . . . . . . . . . . . . . . . . . . . . f ( x ) Figure 16.1. Geometrical interpretation of MVT. This Lagrange’s mean value theorem is an important theorem in differentiable calculus. This theorem was first discovered by Joseph Louis Lagrange (1736–1813), but the idea of applying the Rolle’s theorem to a suitably contrived auxiliary function was given by Ossian Bonnet (1819– 1892). However, the first statement of the theorem appears in a paper by renowned physicist Andr´ e-Marie Amp´ ere (1775–1836).

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  Mathematical Analysis Explained

  • Neil A Watson(Author)
  • 1993(Publication Date)
  • WSPC

    (Publisher)

  Mean value theorems 65 Theorem 4.7 (Cauchy's mean value theorem.) Suppose that f and g are continuous on [a, b] and differentiable on ]a, b[, and that g f (x) is never zero. Then there is c G]a, b[ such that f(b) ~ /(a) = fM , 4fi * g(b)-g(a) g'(c) ' { ' } Cauchy's mean value theorem appears to be more general than L a -grange's, since the latter can be obtained by taking g(x) = x in the former. However, since g'(x) is never zero it is either always positive or always negative (by Theorem 4.8 below), so that g is invertible, and Cauchy's the-orem can be obtained by applying Lagrange's to / o g~ l on [g(a),g(b)} (if g is strictly increasing) or [g(b),g(a) (if g is strictly decreasing); see Exercise 4.22. The proof is similar to that of Lagrange's theorem, the auxiliary function (j) being given by / — kg for a suitably chosen constant k. But there is a bit more to it, since we first need to show that the left-hand side of (4.8) is defined; that is, that g(b) - g(a) ^ 0. This follows from Rolle's theorem, since if g(a) = g(b) then g'{d) = 0 for some d G]a,6[, contrary to our hypothesis that g'(x) is never zero. Note that we cannot just apply Lagrange's theorem to / and g separately to prove Cauchy's, as the two values of c we obtained would generally be different. Proof. If g(b) = g(a) then, by Rolle's theorem, there would be a point in ]a, b[ where g' was zero, contrary to our hypothesis. Hence g(b) —g(a) ^ 0. Put (&); that is, take h = f(b)-m g(b)-g(a) ' Then satisfies all the hypotheses of Rolle's theorem, so that there is c e]a, b[ such that (f>'(c) = 0, and hence m = . = f(b)-f(a) g'(c) g(b)-g(a)- 66 DIFFERENTIABLE FUNCTIONS The intermediate value property for derivatives Although a derivative need not be continuous (see Exercise 4.7), we can show that it must have the intermediate value property.

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  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  (Hint: Consider the function f (x) = sin 2x on the interval [s, t] and follow the lead of Example 5.) 38. Use the MVT to show that √ 1 + t ≤ t 2 + 1 for t ≥ 0. (Hint: Apply the MVT to the function f (x) = √ 1 + x on the interval [0, t].) 39. Let f (x) = |x|. a) Find the average rate of change of f on [-1, 2]. b) Show that there is no number c on (-1, 2) for which f 0 (c) is equal to the average rate of change found in part a. c) Does the answer to part b contradict The Mean Value Theorem? Explain. 40. Let f (x) = x 2 3 . a) Find f 0 (x). b) Does f satisfy the conditions for The Mean Value Theorem on the interval (-1, 27)? c) Find a number c in (-1, 27) such that the instanta- neous rate of change of f at c is equal to the average rate of change of f on (-1, 27). d) Does your answer to part c contradict The Mean Value Theorem? Explain. Applications 41. Velocity A ball thrown up into the air has height in feet after t seconds given by s(t) = -16t 2 + 32t + 5. a) Find the average velocity of the ball from t = 1 to t = 2. b) Find the time at which the instantaneous velocity of the ball is the same as the average velocity found in part a. 42. Water Flow During a 3-minute time interval, the flow of water into a tank is regulated in such a way that the volume after t minutes is V (t) = 0.1t 3 + 2t + 5 gallons. a) Find the average rate of flow of the water over the 3-minute time interval. b) Find the time at which the instantaneous flow rate of the water is the same as the average rate of flow found in part a. 43. Marginal and Average Costs In economics, C(x) repre- sents the cost of producing x units, C(x) = C(x) x is the average cost per unit, and C 0 (x) is the marginal cost.

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  Advanced Calculus

  An Introduction to Linear Analysis

  • Leonard F. Richardson(Author)
  • 2011(Publication Date)
  • Wiley-Interscience

    (Publisher)

  (See Fig. 4.2.) Observe that this is a generalization of Rolle's Theorem. Proof: Observe that the straight line that is the chord joining the endpoints of the graph of f is described by the equation f(b) -f(a) y = f(a) + (x -a). b -a Now, for all x E [a, b], let f(b) -f(a) h(x) = f(x)-y = f(x)-f(a)-b -a (x-a) , The Mean Value Theorem 107 [h] Figure 4.2 f (x) = ~, chord with parallel tangent. the difference in height between the graph of f and the graph of the chord. Observe that hE C[a, b] and h(a) = 0 = h(b). Also, hi exists at least on (a, b). Thus Rolle 's Theorem implies there exists p, E (a, b) such that hi (p,) = O. This implies that • Corollary 4.2.1 If fl == 0 on an interval I, then f is a constant function on I. Proof: Fix any a E I and let x E I be arbitrary. Then The Mean Value Theorem implies there exists p, E I such that f(x) -f(a) = f'(p,) (x -a) = O(x -a) = 0 so that f (x) == f (a), a constant function. • Corollary 4.2.2 If l' == gl on an interval I, then there exists c E lR such that f(x) == g(x) + c. Proof: Let h(x) = f(x) -g(x) and observe that hi == 0 on I. Now use Corollary 4.2.1. • Corollary 4.2.3 Suppose fEe [a, b] is differentiable at least on (a, b). Then 108 THE DERIVATIVE i. If f'(x) 2': Of or all x E (a, b), then f is increasing (denoted by f /) on [a, b]. ll . If f' (x) > 0 for all x E (a, b), then f is strictly increasing (denoted by f r) on [a , b]. iii. Iff' (x) ::; 0 for all x E (a, b), then f is decreasing (denoted by f -.,,) on [a, b ]. iv. If f' (x) < 0 for all x E (a, b), then f is strictly decreasing (denoted by f 1) on [a , b ]. Proof: i. Suppose that f'(x) 2': 0 for all x E (a, b). Then if Xl < X2 are points of [a, b], The Mean Value Theorem implies there exists fJ E (a, b) such that so that f(X2) 2': f(XI), and f is increasing on [a, b]. For the remaining cases, see Exercises 4.10-4.12. • EXAMPLE 4.2 If x> 0, then s in x < x. Proof: Let f( x) = x -sinx. Then f(O) = 0 and f'( x) = 1 -cosx > 0 on (0, 27f).

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  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Find that order size. Mean Value Theorem In Exercises 35 and 36, copy the graph and sketch the secant line to the graph through the points parenleft.alt1a, f parenleft.alt1aparenright.alt1parenright.alt1 and parenleft.alt1b, f parenleft.alt1bparenright.alt1parenright.alt1. Then sketch any tangent lines to the graph for each value of c guaranteed by The Mean Value Theorem. To print an enlarged copy of the graph, go to MathGraphs.com. 35. x a b f y 36. x a b f y Writing In Exercises 37–40, explain why The Mean Value Theorem does not apply to the function f on the interval [0, 6]. 37. y x f 1 2 3 4 5 6 1 2 5 6 3 4 38. y x 1 2 3 4 5 6 1 2 5 6 3 4 f 39. f (x) = 1 x - 3 40. f (x) = uni2223 x - 3 uni2223 4.2 Rolle’s Theorem and The Mean Value Theorem 219 41. Mean Value Theorem Consider the graph of the function f (x) = -x 2 + 5 (see figure). (a) Find the equation of the secant line joining the points (-1, 4) and (2, 1). (b) Use The Mean Value Theorem to determine a point c in the interval (-1, 2) such that the tangent line at c is parallel to the secant line. (c) Find the equation of the tangent line through c. (d) Use a graphing utility to graph f, the secant line, and the tangent line. - 4 2 4 - 2 2 6 (- 1, 4) (2, 1) x y f (x) = - x 2 + 5 (- 2, - 6) (4, 0) x y f (x) = x 2 - x - 12 - 4 - 8 8 - 12 Figure for 41 Figure for 42 42. Mean Value Theorem Consider the graph of the function f (x) = x 2 - x - 12 (see figure). (a) Find the equation of the secant line joining the points (-2, -6) and (4, 0). (b) Use The Mean Value Theorem to determine a point c in the interval (-2, 4) such that the tangent line at c is parallel to the secant line. (c) Find the equation of the tangent line through c. (d) Use a graphing utility to graph f, the secant line, and the tangent line. Using The Mean Value Theorem In Exercises 43– 56, determine whether The Mean Value Theorem can be applied to f on the closed interval [a, b].

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