(D) Define a function $h(x) = g(x) - 4f(x)$ on the interval $[2, 4]$.Since $f(x)$ and $g(x)$ are differentiable,$h(x)$ is also differentiable on $(2,

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$g'(x) = 4f'(x) \text{ for at least one } x \in (2, 4)$

(D) Define a function $h(x) = g(x) - 4f(x)$ on the interval $[2, 4]$.Since $f(x)$ and $g(x)$ are differentiable,$h(x)$ is also differentiable on $(2, 4)$ and continuous on $[2, 4]$.Calculate the values at the endpoints:$h(2) = g(2) - 4f(2) = 0 - 4(8) = -32$.$h(4) = g(4) - 4f(4) = 8 - 4(10) = 8 - 40 = -32$.Since $h(2) = h(4)$,by Rolle's Theorem,there exists at least one $c \in (2, 4)$ such that $h'(c) = 0$.Since $h'(x) = g'(x) - 4f'(x)$,we have $g'(c) - 4f'(c) = 0$,which implies $g'(c) = 4f'(c)$ for at least one $c \in (2, 4)$.Thus,option $D$ is correct.

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

Let $f(x)$ and $g(x)$ be two differentiable functions in $R$ such that $f(2) = 8, g(2) = 0, f(4) = 10$,and $g(4) = 8$. Then which of the following is true?

A
$g'(x) > 4f'(x) \text{ for all } x \in (2, 4)$
B
$3g'(x) = 4f'(x) \text{ for at least one } x \in (2, 4)$
C
$g(x) > f(x) \text{ for all } x \in (2, 4)$
D
$g'(x) = 4f'(x) \text{ for at least one } x \in (2, 4)$

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

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MediumAIEEE 2005

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If Rolle's theorem holds for the function $f(x) = 2x^3 + bx^2 + cx$ on the interval $x \in [-1, 1]$ at the point $x = \frac{1}{2}$,then the value of $2b + c$ is:

MediumJEE MAIN 2015

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Let $f$ be a function which is continuous and differentiable for all $x$. If $f(1) = 1$ and $f^{\prime}(x) \leq 5$ for all $x$ in $[1, 5]$,then the maximum value of $f(5)$ is

MediumMHT CET 2025

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Consider the quadratic equation $ax^2 + bx + c = 0$ where $2a + 3b + 6c = 0$ and let $g(x) = a \frac{x^3}{3} + b \frac{x^2}{2} + cx$.
Statement-$1$: The quadratic equation has at least one root in the interval $(0, 1)$.
Statement-$2$: Rolle's theorem can be applied to the function $g(x)$ in the interval $[0, 1]$.

Difficult

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Let the function $f:[-7,0] \rightarrow R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f'(x) \leq 2$ for all $x \in (-7,0)$,then for all such functions $f$,$f(-1)+f(0)$ lies in the interval:

DifficultJEE MAIN 2020

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Let $f(x)$ and $g(x)$ be two differentiable functions in $R$ such that $f(2) = 8, g(2) = 0, f(4) = 10$,and $g(4) = 8$. Then which of the following is true?

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If the equation $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x = 0$,where $a_1 \neq 0$ and $n \ge 2$,has a positive root $x = \alpha$,then the equation $n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \dots + a_1 = 0$ has a positive root,which is:

If Rolle's theorem holds for the function $f(x) = 2x^3 + bx^2 + cx$ on the interval $x \in [-1, 1]$ at the point $x = \frac{1}{2}$,then the value of $2b + c$ is:

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