Question

Suppose that $f\left(0\right)=-3$ and $f^{\prime }\left(x\right)\le 5$ for all values of $x$. Then the largest value which $f\left(2\right)$ can attain is

• A. $7$
• B. $-7$
• C. $13$
• D. $8$

Answer 1

The correct option is A $7$

By Lagrange's mean value theorem, there exists $c\in (0,2)$ such that
$f^{\prime }\left(c\right)=\frac{f\left(2\right)-f\left(0\right)}{2}$
$f^{\prime }\left(c\right)\le 5$ [$\because f^{\prime }\left(x\right)\le 5$ for all $x$]
$\Rightarrow f\left(2\right)+3\le 10$
$\Rightarrow f\left(2\right)\le 7$