Question

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

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

Answer 1

The correct option is C

$28$

Explanation for the correct option.

Find the maximum value of $f\left(7\right)$.

If a function is differentiable in the interval $\left[a,b\right]$ then for a value $c$ in the interval $f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

Now it is given that the function is differentiable in the interval $\left[2,7\right]$ and $f\text{'}\left(x\right)\le 5$, so

$$\begin{gathered} f\text{'}\left(x\right)\le 5 \\ \Rightarrow \frac{f\left(7\right)-f\left(2\right)}{7-2}\le 5 \\ \Rightarrow f\left(7\right)-3\le 5\times 5\left[f\left(2\right)=3\right] \\ \Rightarrow f\left(7\right)\le 25+3 \\ \Rightarrow f\left(7\right)\le 28 \end{gathered}$$

Thus the maximum possible value of $f\left(x\right)$ at $x=7$ is $28$.

Hence, the correct option is C.