Question

lf f is differentiable for all $x,f^{\prime }\left(x\right)\ge 3\forall x$ in [2, 7] and $f\left(7\right)=19$, then $f\left(2\right)$ less than or equal to

Answer 1

Given $f\left(x\right)$ is differentiable for all $x$.
$\Rightarrow$ $f\left(x\right)$ is continuous for all $x$
So, we can say $f\left(x\right)$ is continuous on $[2,7]$ and differentiable on $(2,7)$, then there is $c\in (2,7)$ such that
$f^{\prime }\left(c\right)=\frac{f\left(7\right)-f\left(2\right)}{5}$
$\Rightarrow f^{\prime }\left(c\right)=\frac{19-f\left(2\right)}{5}$ ...(1)
Also, given $f^{\prime }\left(x\right)\ge 3\forall x\in [2,7]$
$\Rightarrow f^{\prime }\left(c\right)\ge 3$
$\Rightarrow \frac{19-f\left(2\right)}{5}\ge 3$
$\Rightarrow 19-f\left(2\right)\ge 15$
$\Rightarrow f\left(2\right)\le 4$