Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 199: 61

See explanations.

Step 1. Given $f'(x)\leq1$ for $1\leq x\leq4$, based on the Mean Value Theorem, there is a point $c$ in the interval such that $f'(c)=\frac{f(4)-f(1)}{4-1}$ or $f(4)-f(1)=3f'(c)$ Step 2. Since the condition $f'(x)\leq1$ is true for any points in the interval, we have $f'(c)\leq1$, and $f(4)-f(1)=3f'(c)\leq3$