Question

If $f\left(x\right)$ is continuous and differentiable over $[-2,5]$ and $-4\le f^{\prime }\left(x\right)\le 3$ for all $x$ in $(-2,5),$ then the greatest possible value of $f\left(5\right)-f(-2)$ is

• A. $7$
• B. $9$
• C. $15$
• D. $21$

Answer 1

The correct option is D $21$

Given that $-4\le f^{\prime }\left(x\right)\le 3$
Using LMVT on $f\left(x\right)$ in $[-2,5]$, we get
$-4\le \frac{f\left(5\right)-f(-2)}{5-(-2)}\le 3$
$\Rightarrow -28\le f\left(5\right)-f(-2)\le 21$
So, the greatest possible value of $f\left(5\right)-f(-2)$ is $21.$