(C) According to the Lagrange's Mean Value Theorem $(LMVT)$, for a function $f$ continuous on $[2, 4]$ and differentiable on $(2, 4)$, there exists at

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(C) According to the Lagrange's Mean Value Theorem $(LMVT)$, for a function $f$ continuous on $[2, 4]$ and differentiable on $(2, 4)$, there exists at least one $c \in (2, 4)$ such that $f^{\prime}(c) = \frac{f(4) - f(2)}{4 - 2}$.Given that $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$, we have $f^{\prime}(c) \geq 6$.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 which is differentiable for all real $x$. If $f(2) = -4$ and $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$, then which of the following is true?

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A
$f(4) < 8$
B
$f(4) \geq 12$
C
$f(4) \geq 8$
D
$f(4) < 12$

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Rolle’s theorem, Lagrange's mean value theorem

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

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