1) It's given that $f(x)$ is differentiabile function, so it implies $f(x)$
is continuous function, the hypothesis for mean value was
satisfied, according to Lagrange's mvt if function $f(x)$ is
continuous on interval $[a,b]$ and differentiable on $(a,b)$ then
there exists point c in $(a,b)$ such that $f'(c) = \frac{f(b) -f(a)}{b-a}$
Now apply LMVT on interval [1,5] it gives us there exists point c in
$(1,5)$ such that $f'(c) = 4$, which makes the statement false.
2) Similar approach as above, it gives us there is c in (1,3) with
f'(c)= 1, which makes the statement 2 false.
3)Not necessarily, function can be discontinuous.
4)which is very obvious, so it's true.
