real analysis - Using mean value theorem and Rolle's theorem to prove an inequality

$f$ is a continuous function on
${[}0,+\infty{)}$ and differentiable on ${(}0,+\infty{)}$, and $f$ is strictly decreasing on ${(}0,+\infty{)}$

the question is: is this inequality true for each $(a,b)$ $\in ({[}0,+\infty{)})^2$ such as $a

$(b-a)f(b)\leqslant f(b)-f(a) \leqslant (b-a)f(a)$

I've tried to prove that $\forall x \in {[}a,b{]}$ : $f(b)\leqslant f'(x)\leqslant f(a)$ so I can apply the mean value theorem
on ${[}a,b{]}$