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Theorem (2.21 Mean Value Theorem): Let f [a, b] = R be continuous and differentiable on (a, b). Then there exists (at least one) x â‚¬ (a, b) such that f(b) – f(a) = f'(x)(b – a).

Proof: If L is defined to be the function whose graph is the straight line connecting the points (a, f(a)) and (b, f(b)), then f(b) – f(a) = L'(x)(b – a) for all x. Define g(x) = f(x) – L(x) for x â‚¬ [a, b]. Then g is continuous on [a, b] and differentiable on (a, b). Moreover, g(a) = g(b) = 0, so g'(x) = 0 for some x â‚¬ (a, b) by Rolle’s Theorem. For this x, f'(x) = L'(x) = (f(b) – f(a))/(b – a).

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03:58

Let f and g be continuous functions on [a, b] and differentiable at every point in the interior, with g(a) â‰ g(b). Prove that there exists a point c in (a, b) such thatf'(c) = (f(b) – f(a)) / (g(b) – g(a)) * g'(c)

(Hint: apply the mean value theorem to the function(f(b) – f(a))g(c) – (g(b) – g(a))f(c))

This is sometimes called the second mean value theorem.

02:24

If f(x) is continuous on [a,b] and differentiable on (a,b), then the Mean Value Theorem states that there is a point between a and b such that f'(c) = (f(b) – f(a))/(b – a).

01:03

Use Rolle’s Theorem to prove the slightly more general theorem from Exercise 68 : If $f$ is continuous on $[a, b]$ and differentiable on $(a, b),$ and if $f(a)=f(b),$ then there is some value $c \in(a, b)$ with $f^{\prime}(c)=0 .$ (Hint: Apply Rolle’s Theorem to the function $g(x)=f(x)-f(a) .)$

02:32

Rolle’s Theorem: Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one number c in (a, b) such that f'(c) = 0.

08:36

Use Rolle’s Theorem to prove the Mean Value Theorem. Suppose that $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b) .$ Let $g(x)$ be the difference between $f(x)$ and the $y$ -value on the secant line joining $(a, f(a))$ to$(b, f(b)),$ so$$g(x)=f(x)-f(a)-\frac{f(b)-f(a)}{b-a}(x-a)$$(a) Show $g(x)$ on a sketch of $f(x)$(b) Use Rolle’s Theorem (Problem 44 ) to show that there must be a point $c$ in $(a, b)$ such that $g^{\prime}(c)=0$(c) Show that if $c$ is the point in part (b), then$$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$$

Transcript

For the given question, we need to prove mean value. Now, here we are given that let f be a function defined on interval a ,b to R be a continuous and be a differentiable function on open interval a ,b and it is continuous on closed interval a ,b then there exist some x which belongs to a ,b such that f dash of x is equal to f of b minus f of a divided by b minus a. So here further let I be defined as a function of a graph whose straight line is connecting the point a ,f ,f and b ,f ,b. So here let L be the line connecting point which is a ,f of a and b ,f of b. So here we can say that L dash of x is equal to f of b minus f of a divided by b minus a. So here further on simplifying this we can say that we have proved that here this is…

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