(C) Given that $f(x)$ is differentiable on $[2, 5]$.By Lagrange's Mean Value Theorem $(LMVT)$,there exists at least one $c \in (2, 5)$ such that $f'(c

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(C) Given that $f(x)$ is differentiable on $[2, 5]$.By Lagrange's Mean Value Theorem $(LMVT)$,there exists at least one $c \in (2, 5)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.Here,$a = 2$ and $b = 5$.$f'(c) = \frac{f(5) - f(2)}{5 - 2} = \frac{1/2 - 1/5}{3}$.$f'(c) = \frac{(5 - 2)/10}{3} = \frac{3/10}{3} = \frac{1}{10}$.

Class 12 Mathematics Continuity and Differentiation Rolle’s theorem, Lagrange's mean value theorem

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If $f(x)$ is differentiable on the interval $[2, 5]$ such that $f(2) = 1/5$ and $f(5) = 1/2$,then there exists a number $c$ such that $2 < c < 5$ and $f'(c) = \dots$

A
$1/2$
B
$1/5$
C
$1/10$
D
None of these

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

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If $f(x)$ is differentiable on the interval $[2, 5]$ such that $f(2) = 1/5$ and $f(5) = 1/2$,then there exists a number $c$ such that $2 < c < 5$ and $f'(c) = \dots$

Similar Questions

If $2a + 3b + 6c = 0$ and $a, b, c \in \mathbb{R}$,then the equation $ax^2 + bx + c = 0$ has at least one root between $0$ and $1$.

Suppose $f(x)$ is twice differentiable in the interval $[1, 3]$ and $f(1)=f(3)$. If $|f^{\prime \prime}(x)| \leq 2$,then for all $x$ in $[1, 3]$,which one of the following is true?

The number of values of $C$ that satisfy the conclusion of Rolle's theorem for the function $f(x) = \sin(2 \pi x)$ on the interval $x \in [-1, 1]$ is:

Consider the function $f(x) = |x - 2| + |x - 5|, x \in R$.Statement-$1$: $f'(4) = 0$.Statement-$2$: $f$ is continuous in $[2, 5]$,differentiable in $(2, 5)$ and $f(2) = f(5)$.

The value of $c$ satisfied by Rolle's theorem for the function $f(x) = x^2(1 - x)^2$ on the interval $x \in [0, 1]$ is...

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Consider the function $f(x) = |x - 2| + |x - 5|, x \in R$.
Statement-$1$: $f'(4) = 0$.
Statement-$2$: $f$ is continuous in $[2, 5]$,differentiable in $(2, 5)$ and $f(2) = f(5)$.