Question

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

• A. $7$
• B. $15$
• C. $28$
• D. $14$

Answer 1

Answer: (C)

$28$

Given,

$f^{\prime }\left(x\right)\le 5$
$\therefore \frac{f\left(7\right)-f\left(2\right)}{x-2}\le 5$ [by Lagrange mean value theorem]

$\Rightarrow \frac{f\left(7\right)-3}{7-2}\le 5$

$\Rightarrow f\left(7\right)\le 5(7-2)+3$

$\therefore f\left(7\right)\le 28$