Find a max possible value of an undefined function in an interval when derivative is a constant

Question

The function $f(x)$ is continuous and differentiable in $[0,1]$ if $f'(x)\le 10$ for all $x\in[0,1]$ and $f(0)=0$,

What is the maximum possible value of $f(x)$ for $x\in [0,1]$ ?

Any help would be greatly appreciated, thanks.

Answer 1

$\forall x\in [0,1]$ we have,

$\frac{f(x)-f(0)}{x-0}\le 10$ {By lagrange's mean value theorem we have a $c\in [0,x]\ such\ that\ \frac{f(x)-f(0)}{x-0}=f'(c)\le10$}

$\therefore$ maximum value of f(x) is 10$x$ in [0,1].

Hope it helps:)