Question

Let $f\left(x\right)$ be a continuous and differentiable function on the interval $[2,10]$. It is known that $f\left(2\right)=8$ and the derivative on the given interval satisfies the condition $f^{\prime }\left(x\right)\le 4$ for all $x\in (2,10)$. Then the maximum possible value of the function at $x=10$ is

• A. $40$
• B. $32$
• C. $35$
• D. $30$

Answer 1

The correct option is A $40$

Using Lagrange's formula, we have:
$f^{\prime }\left(c\right)=\frac{f\left(10\right)-f\left(2\right)}{10-2}$
where $c$ is a point lying in the interval $(2,10)$
$\Rightarrow f^{\prime }\left(c\right)\le 4$
$\Rightarrow \frac{f\left(10\right)-f\left(2\right)}{10-2}\le 4$
$\Rightarrow f\left(10\right)\le 40$
$\therefore f(10{)}_{max}=40$