Question

If $f$ is a differentiable function for all real $x$ and $f^{'}\left(x\right)\leq 5\forall x\in R$ . If $f\left(2\right)=0$ and $f\left(5\right)=15$ value of $f\left(3\right)$ is

Answer 1

$\frac{f \left(3\right) - f \left(2\right)}{3 - 2}=f^{'}\left(4\right)$
$f\left(3\right)\leq 5...\left(1\right)$
$\frac{f \left(5\right) - f \left(3\right)}{5 - 3}=f^{'}\left(2\right)$
$f\left(3\right)\geq 5...\left(2\right)$
Hence $f\left(3\right)=5$