Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 199: 67

See explanations.

Step 1. The given conditions indicate that the Mean Value Theorem can be applied to the given $f'(c)=\frac{f(b)-f(a)}{b-a}$ where $c$ is a point in the interval $(a,b)$. Step 2. Since $f(b)\lt f(a)$ and $b\gt a$, we have $f'(c)\lt0$