(B) Given that $f(x)$ is continuous on $[2, 4]$ and differentiable on $(2, 4)$.By Lagrange's Mean Value Theorem,there exists at least one $c \in (2, 4

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(B) Given that $f(x)$ is continuous on $[2, 4]$ and differentiable on $(2, 4)$.By Lagrange's Mean Value Theorem,there exists at least one $c \in (2, 4)$ such that $f'(c) = \frac{f(4) - f(2)}{4 - 2}$.We are given $f'(x) \geq 6$ for all $x \in [2, 4]$,so $f'(c) \geq 6$.Substituting the values,we get $\frac{f(4) - (-4)}{2} \geq 6$.$f(4) + 4 \geq 12$.$f(4) \geq 8$.

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

Medium

Let $f$ be a function that is continuous and differentiable for all real $x$. If $f(2) = -4$ and $f'(x) \geq 6$ for all $x \in [2, 4]$,then which of the following is true?

A
$f(4) < 8$
B
$f(4) \geq 8$
C
$f(4) \geq 12$
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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Let $f$ be a function that is continuous and differentiable for all real $x$. If $f(2) = -4$ and $f'(x) \geq 6$ for all $x \in [2, 4]$,then which of the following is true?

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