(B) Let $h(x) = f(x) - 2g(x)$.Since $f$ and $g$ are differentiable on $[0, 1]$,$h(x)$ is also differentiable on $[0, 1]$.Calculate the values of $h(x)

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(B) Let $h(x) = f(x) - 2g(x)$.Since $f$ and $g$ are differentiable on $[0, 1]$,$h(x)$ is also differentiable on $[0, 1]$.Calculate the values of $h(x)$ at the endpoints:$h(0) = f(0) - 2g(0) = 2 - 2(0) = 2$.$h(1) = f(1) - 2g(1) = 6 - 2(2) = 6 - 4 = 2$.Since $h(0) = h(1) = 2$,by Rolle's Theorem,there exists at least one $c \in (0, 1)$ such that $h'(c) = 0$.$h'(x) = f'(x) - 2g'(x)$.Setting $h'(c) = 0$ gives $f'(c) - 2g'(c) = 0$,which implies $f'(c) = 2g'(c)$.

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

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If $f$ and $g$ are differentiable functions in $[0, 1]$ satisfying $f(0) = 2$,$g(1) = 2$,$g(0) = 0$,and $f(1) = 6$,then for some $c \in (0, 1)$:

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A
$f'(c) = g'(c)$
B
$f'(c) = 2g'(c)$
C
$2f'(c) = g'(c)$
D
$2f'(c) = 3g'(c)$

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

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MediumWBJEE 2014

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If $f$ and $g$ are differentiable functions in $[0, 1]$ satisfying $f(0) = 2$,$g(1) = 2$,$g(0) = 0$,and $f(1) = 6$,then for some $c \in (0, 1)$:

Similar Questions

If $f(x) = ax^3 + bx^2 + 11x - 6$ for $x \in [1, 3]$ satisfies the conditions of Rolle's theorem and $f'\left( 2 + \frac{1}{\sqrt{3}} \right) = 0$,find $a$ and $b$.

Let $f(x)$ be a differentiable function in $[2,7]$. If $f(2)=3$ and $f^{\prime}(x) \leq 5$ for all $x$ in $(2,7)$, then the maximum possible value of $f(x)$ at $x=7$ is

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