Let f(x) and g(x) be two differentiable funct | vedclass

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

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(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.

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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