Chapter 4: Applications of Derivatives - Section 4.2 - The Mean Value Theorem - Exercises 4.2 - Page 199: 63

See explanations.

Step 1. Consider the function $f(x)=cos(x)$ on $[0,x]$, which satisfies the conditions for the Mean Value Theorem. We have $f'(c)=\frac{f(x)-f(0)}{x-0}=\frac{cos(x)-1}{x}$ for a value $c$ in the interval. Step 2. Rewrite the above result as $|cos(x)-1|=|xf'(c)|$. As $|f'(x)|=|-sin(x)|\leq1$, we have $|cos(x)-1|=|xf'(c)|=|x\cdot sin(c)|\leq|x|$