Question

Suppose f(x) is continuous on [2,8] and −4≤f′(x)≤5 for all x in (2,8). Use the Mean Value Theorem to estimate f(8)−f(2).

Answer 1

by mvt
\[\frac{f(8)-f(2)}{8-2}=f'(c)\] for some c in (2,8)

Answer 2

so you have
\[-4\leq\frac{f(8)-f(2)}{6}\leq 5\]

Answer 3

therefore
\[-24\leq f(8)-f(2)\leq 30\]