Question

Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?

Options

• \[f\] is constant on \[[a,b]\].

• \[f\] is decreasing on \[[a,b]\].

• \[f\] is strictly increasing on \[[a,b]\].

• \[f\] is increasing on \[[a,b]\].

Answer 1

The first derivative test gives decreasing behaviour when \[f'(x)\leq0\] throughout \[(a,b)\]. The derivative may be zero at some points.