Answer
See explanations.
Work Step by Step
Step 1. Identify the given conditions: $f’(x)\leq0$ on $[a,c)$ and $f’(x)\geq0$ on $(c,b]$.
Step 2. Using the sign test for $f’(x)$ across the critical point $c$, we have $..(-)..(c)..(+)..$ and we can see that the curve is concave up with a local minimum at $x=c$, which means that $f(x)\geq c$ on the interval.
