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(D) Given that $f(x)$ is continuous on $[1, 5]$ and differentiable on $(1, 5)$.By the Lagrange's Mean Value Theorem,there exists at least one $c \in (

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$21$

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(D) Given that $f(x)$ is continuous on $[1, 5]$ and differentiable on $(1, 5)$.By the Lagrange's Mean Value Theorem,there exists at least one $c \in (1, 5)$ such that $f^{\prime}(c) = \frac{f(5) - f(1)}{5 - 1}$.We are given $f^{\prime}(x) \leq 5$ for all $x \in [1, 5]$,so $f^{\prime}(c) \leq 5$.Substituting the values,we get $\frac{f(5) - 1}{4} \leq 5$.$f(5) - 1 \leq 20$.$f(5) \leq 21$.Therefore,the maximum value of $f(5)$ is $21$.

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

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