# What is Mean value theorem: Definition and 150 Discussions

In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
More precisely, the theorem states that if

f

{\displaystyle f}
is a continuous function on the closed interval

[
a
,
b
]

{\displaystyle [a,b]}
and differentiable on the open interval

(
a
,
b
)

{\displaystyle (a,b)}
, then there exists a point

c

{\displaystyle c}
in

(
a
,
b
)

{\displaystyle (a,b)}
such that the tangent at

c

{\displaystyle c}
is parallel to the secant line through the endpoints

(
a
,
f
(
a
)
)

{\displaystyle (a,f(a))}
and

(
b
,
f
(
b
)
)

{\displaystyle (b,f(b))}
, that is,

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1. ### I Continuity of Mean Value Theorem

Suppose f:]a,b[\to\mathbb{R} is some differentiable function. Then it is possible to define a new function ]a,b[\to [a,b],\quad x\mapsto \xi_x in such way that f(x) - f(a) = (x - a)f'(\xi_x) for all x\in ]a,b[. Mean Value Theorem says that these \xi_x exist. One question that sometimes...
2. ### I Why Can We Use This Mathematical Formula for Mean Value of Measurement?

Hi, I'm new to the quantum world, and would like to know why mathematically can we say that for mean value of measurment: <T>=<phi|T|phi> ?
3. ### Understanding the Mean value theorem

Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point ##c## and a secant line joining the two end points. Or Theorem only holds for one tangent point. Cheers
4. ### I Mean value theorem - prove inequality

I don't need an answer (although I don't have sadly, it's from a test). I need just a tip on how to start it... i cannot use Taylor in here (##\ln(x)## is not Taylor function), therefore, its only MVT, but I don't know which point I should try... since I must get the annoying ##\ln(x)##...
5. ### A Noether's theorem time invariance -- mean value theorem use?

how does the first step use mean value theorem? I don't get it , can anyone explain , thanks.
6. ### MHB 34 MVT - Application of the mean value theorem

$10 min = \dfrac{h}{6}$ So $a(t)=v'(t) =\dfrac{\dfrac{(50-30)mi}{h}}{\dfrac{h}{6}} =\dfrac{20 mi}{h}\cdot\dfrac{6}{h}=\dfrac{120 mi}{h^2}$ Hopefully 🕶
7. ### MHB Mean Value Theorem: Showing Change in a Function is Bounded

Ok Just have trouble getting this without a function..
8. ### MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines

$\tiny{3.2.15}$ Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$. $f(x)=\sqrt{x} \quad [0,4]$ Are the secant line and the tangent line parallel...
9. ### Mean Value Theorem: Proof & Claim

a) Proof: By theorem above, there exists a ##a \in \mathbb{R}## such that for all ##x \in I## we have ##f'(x) = a##. Let ##x, y \in I##. Then, by Mean Value Theorem, $$a = \frac{f(x) - f(y)}{x - y}$$ This can be rewritten as ##f(x) = ax - ay + f(y)##. Now, let ##g(y) = -ay + f(y)##. Then...
10. ### MHB Function Help - Rolle's Theorem or the Mean Value Theorem?

Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0. (a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0. (b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0. If you use a major theorem, then cite the theorem...
11. ### MHB Understanding Theorem 3.4: Mean Value Theorem & Cauchy Riemann Equations

I am reading "Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ... ... I am focused on Section 3.2 The Cauchy Riemann Equations ... I need help in fully understanding the Proof of Theorem 3.4 ...The start of Theorem 3.4 and its...
12. ### B Thinking about the mean value theorem without geometry

Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
13. ### MHB Show inequality using the mean value theorem

Hey! :o Let $D=\left \{x=(x_1, x_2)\in \mathbb{R}^2: x_2>\frac{1}{x_1}, \ x_1>0\right \}$. We have the function $f: D\rightarrow \left (0,\frac{\pi}{2}\right )$ with $f(x)=\arctan \left (\frac{x_2}{x_1}\right )$. I want to show using the mean value theorem in $\mathbb{R}^2$ that for all...
14. ### Find Point c that satisfies the Mean Value Theorem

Homework Statement Find the point "c" that satisfies the Mean Value Theorem For Derivatives for the function ## f(x) = \frac {x-1} {x+1}## on the interval [4,5]. Answer - c = 4.48 Homework Equations ##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}## ##f'(c) = \frac { f(b) - f(a)} {b-a}## The Attempt...
15. ### Verifying Hypotheses of the Mean Value Theorem for f(x)=1/(x-2) on [1,4]

Homework Statement Find all the numbers c that satisfy the conclusion of the Mean Value Theorem for the functions f(x)=\dfrac{1}{x-2} on the interval [1, 4] f(x)=\dfrac{1}{x-2} on the interval [3, 6] I don't need help solving for c, I just want to know how I can verify that the hypotheses of...
16. ### Error approximation using mean value theorem for mv-function

Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...
17. ### Using the mean value theorem to prove the chain rule

Homework Statement I and J are open subsets of the real line. The function f takes I to J, and the function g take J to R. The functions are in C1. Use the mean value theorem to prove the chain rule. Homework Equations (g o f)' (x) = g'(f (x)) f'(x) MVT The Attempt at a Solution [/B] I know...
18. ### Mean value theorem variation proof

Homework Statement Let f is differentiable function on [0,1] and f^{'}(0)=1,f^{'}(1)=0. Prove that \exists c\in(0,1) : f^{'}(c)=f(c). Homework Equations -Mean Value Theorem The Attempt at a Solution The given statement is not true. Counter-example is f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10...
19. ### Proving the Existence of a Single Real Root Using Derivatives

Homework Statement the original function is ##−6 x^3−3x−2 cosx## ##f′(x)=−2x^2−3+2sin(x)## ##−2x^2 ≤ 0## for all x and ##−3+2 sin(x) ≤ −3+2 = −1##, for all x ⇒ f′(x) ≤ −1 < 0 for all x The Attempt at a Solution this problem is part of a larger problem which says there is a cubic...
20. ### Is this a correct proof of a function's continuity?

Hello, 1. Homework Statement 1) Let f(x) continuous for all x and (f(x)2)=1 for all x. Prove that f(x)=1 for all x or f(x)=-1 for all x. 2) Give an example of a function f(x) s.t. (f(x)2)=1 for all x and it has both positive and negative values. Does it contradict (1) ? 2. The attempt at a...
21. ### Question on Mean Value Theorem & Intermediate Value Theorem

Homework Statement for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]## Homework Equations Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}## The Attempt at a Solution I got the answer for the question but I have made an assumption but I don't know if...
22. ### Question on Mean Value Theorem

Homework Statement Let ###f## be double differentiable function such that ##|f''(x)|\le 1## for all ##x\in [0,1]##. If f(0)=f(1), then, A)##|f(x)|>1## B)##|f(x)|<1## C)##|f'(x)|>1## D)##|f'(x)|<1## Homework Equations MVT: $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ The Attempt at a Solution I first tried...
23. ### Mean Value Theorem/Rolle's Theorem and differentiability

Homework Statement Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1,1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem? Homework EquationsThe Attempt at a Solution f(x) = 1 - x2/3. f(-1) = 1 - 1 = 0 f(1) = 1 - 1 = 0 f' = 2/3 x -1/3. I don't...
24. ### Mean Value Theorem for Definite Integrals

In the MVT for Integrals: ##f(c)(b-a)=\int_a^bf(x)dx##, why does ##f(x)## have to be continuous in ##[a,b]##.
25. ### Mean value theorem, closed intervals

Mean Value Theorem Suppose that ##f## is a function that is continuous on ##[a,b]## and differentiable on ##(a,b)##. Then there is at least one ##c## in ##(a,b)## such that: $$f'(c) = \frac{f(b) - f(a)}{b - a}$$ My question is: wouldn't it be better to state that ##c## is in ##[a,b]## rather...
26. ### Mean Value Theorem for integrals

Homework Statement Prove the Mean Value Theorem for integrals by applying the Mean Value Theorem for derivatives to the function F(x) = \int_a^x \, f(t) \, dt Homework Equations [/B] Mean Value Theorem for integrals: If f is continuous on [a, b], then there exists a number c in [a, b]...
27. ### Mean value theorem for trigonometric function

Homework Statement Verify Lagrange's MVT for ## f(x)= sinx - sin2x ## in [ 0, π ] Homework Equations ## f'(c) = \frac{f(b)-f(a)}{b-a} ## The Attempt at a Solution Got on solving cosx= 2cos2x How to find c lies in [0, π ]? Solved it using quadratic equation but it gives a complicated value...
28. ### Lagrange's mean value theorem problem

Homework Statement Homework Equations Lagrange's mean value theorem The Attempt at a Solution Applying LMVT, There exists c belonging to (0,1) which satisfies f'(c) = f(1)-f(0)/1 = -f(0) But this gets me nowhere close to the options... :(
29. ### Real Analysis - Mean Value Theorem Application

Homework Statement Let f: R -> R be a function such that \lim_{z\to 0^+} zf(z) \gt 0 Prove that there is no function g(x) such that g'(x) = f(x) for all x in R. Homework Equations Supposed to use the mean value theorem. If f(x) is continuous on [a,b] and differentiable on (a,b) then...
30. ### MHB Proving inequality using Mean Value Theorem

Need help with this exercise been stuck on it for a while i think i get the gist of what i am supposed to do but can't seem to get it to work i am definitely missing something. I set h(x)= x-2/3 - g(x) and tried using the mean value theorem on [a,b] and then tried finding the minimum value of...
31. ### What is the mean value theorem

Definition/Summary The mean value theorem states that if a real-valued function f is continuous and differentiable on an open interval (a,b), then there is a point c in that interval such that f'(c) \ =\ (f(b) - f(a))/(b - a). It also applies if the condition of differentiability is...
32. ### MHB Second mean value theorem in Bonnet's form

Using second mean value theorem in Bonnet's form show that there exists a p in [a,b] such that \int_a^b e^{-x}cos x dx =sin ~p I know the theorem but how to show this using that theorem .
33. ### Mean Value Theorem: Homework Solution

Homework Statement http://i.minus.com/jX32eXvLm6FGu.png Homework Equations The MVT applies if 1) The function is continuous on the closed interval [a,b] such that a<b. 2) The function is differentiable on the open interval (a,b) And if the above two conditions are fulfilled...
34. ### Using Mean Value Theorem to Prove a Summation Equation?

Homework Statement Use the mean value theorem to show that \frac{b^3-a^3}{b-a} = \sum_{j=1}^{n}d_j^2 (x_j - x_{j-1}) \text{where} x_{j-1} < d_j < x_j . Homework Equations The mean value theorem states that if f is continuous on [a,b] and differentiable on (a,b) then there exists a c in...
35. ### Maximizing f(x) with Mean Value Theorem

Homework Statement Suppose that f(0)=-3 and f'(x)<=5 for all values of x. The the largest value of f(2) is A)7 B)-7 C)13 D)8Homework Equations The Attempt at a Solution The problem can be easily solved using the mean value theorem but solving it in a different way doesn't give the right answer...
36. ### MHB Mean value theorem for integrals

Hello all, I have a couple of questions. First, about the mean value theorem for integrals. I don't get it. The theorem say that if f(x) is continuous in [a,b] then there exist a point c in [a,b] such that $\int_{a}^{b}f(x)dx=f(c)\cdot (b-a)$ Now, I understand what it means (I think), but...
37. ### Generalization of Mean Value Theorem for Integrals Needed

Hi all, I'm having trouble finding a certain generalization of the mean value theorem for integrals. I think my conjecture is true, but I haven't been able to prove it - so maybe it isn't. Is the following true? If F: U \subset \mathbb{R}^{n+1} \rightarrow W \subset \mathbb{R}^{n}...
38. ### Mean Value Theorem: Find Point [1,4]

Homework Statement State the Mean Value Theorem and find a point which satisfies the conclusions of the Mean Value Theorem for f(x)=(x-1)3 on the interval [1,4]. 2. The attempt at a solution Mean Value Theorem:states that there exists a c∈(a,b) such that f'(c)=\frac{f(b)-f(a)}{b-a}...
39. ### Use the generalised mean value theorem to prove this

Homework Statement Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Using the generalised mean value theorem, prove that: f(x)=f(c) + (x-c)f'(c)+\frac{(x-c)^2}{2}f''(\theta) for some \theta \in (c, x) Homework Equations Hints given suggest consdiering F(x) =...
40. ### MHB Nikki's question at Yahoo Answers regarding the Mean Value Theorem

Here is the question: Here is a link to the question: Calculus 1 Help on Mean Value Theorem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
41. ### The mean value theorem for integrals and Maple

Homework Statement I have function f which is defined upon an interval [a,b]. I have calculated the mean value using the theorem \frac{1}{b-a} \int_{a}^b f(x) dx What I would like to do is to plot in Maple the mean value rectangle. Where the hight of this rectangle represents the mean value...
42. ### What is the Mean Value Theorem Inequality for the Interval [0,1]?

Homework Statement For every x in the interval [0,1] show that:j \frac{1}{4}x+1\leq\sqrt[3]{1+x}\leq\frac{1}{3}x+1 The Attempt at a Solution Well i subtracted 1 from all sides and divided by x and I got: \frac{1}{4}\leq\frac{\sqrt[3]{1+x}-1}{x}\leq\frac{1}{3} But now I need to find a...
43. ### Prove Mean Value Theorem: f(x) on I=(a,b)

Homework Statement 1) Let f be a function differentiable two times on the open interval I and a and b two numbers in I Prove that: \exists c\in]a,b[:\frac{f(b)-f(a)}{b-a}=f'(a)+\frac{b+a}{c}f''(c) 2) Let f be a function differentiable three times on the open interval I and a and b two...
44. ### Application of the mean value theorem.

Homework Statement Let f and g be two continuous functions on [a,b] and differentiable on ]a,b[ such that for every x in ]a,b[ : f'(x)<g'(x) Homework Equations Show that f(b)-f(a)<g(b)-g(a) The Attempt at a Solution So I said that there exists a c in ]a,b[ such that f'(c)=(f(b)-f(a))/(b-a)...
45. ### Langrange's Mean Value Theorem question

Homework Statement Find a function f on [-1,1] such that :- (a) there exists c \in (-1,1) such that f'(c) = 0 and (b) f(a) \neq f(b) for any a\neq b \in [-1,1] Homework Equations Lagrange's Mean Value Theorem (LMVT), which states that if f:[a,b]-->ℝ is a function which is...
46. ### Real Analysis Mean Value Theorem Proof

Homework Statement Let f: R->R be a function which satisfied f(0)=0 and |df/dx|≤ M. Prove that |f(x)|≤ M*|x|. Homework Equations Mean value theorem says that if f is continuous on [a,b] and differentiable on (a,b), then there is a point c such that f'(c)=[f(b)-f(a)]/(b-a). The...
47. ### Prove a limit using the mean value theorem

I am supposed to use the mean-value theorem to show that lim_x→infty(√(x+5)-√(x))=0. Can anyone help me solving this problem? I have tried to set up the mean value theorem, but i just do not know how to proceed.
48. ### F(n) Limit Calculation and Inequality Proof Help

Given f(n) = (1 - (1/n))n I calculate that the limit as n -> infinity is 1/e. Also given that x/(1-x) > -log(1-x) > x with 0<x<1 (I proved this in an earlier part of the question) I want to show that: 1 > (f(60)/f(infinity)) > e-1/59 > 58/59 I have tried using my value for f...
49. ### Using the Mean Value Theorem to Solve for M in f(b)-f(a)/b-a=MHomework Equations

Homework Statement Let f(x)=log(x)+sin(x) on the positive real line. Use the mean value theorem to assure that for all M>0, there exists positive numbers a and b such that f(b)-f(a)/b-a=MHomework Equations f'(x)=1/x+cos(x) The Attempt at a Solution I know that as x→0, f'(x) gets arbitrarily...
50. ### Why is a New Function Created in the Proof of the Mean Value Theorem?

I am reading the proof for the M.V.T, mostly understanding it all, except for this one step. Here is the link to it: http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx#Extras_DerAppPf_MVT It's near the bottom of the page. What I don't precisely is why they create a new...