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

2≤x≤8 −4≤f′(x)≤5 f is continuous in differentable on [2,8] so we know there is c btw 2 and 8 such that f′(c)=f(8)−f(2)/8−2 And we know −4≤f′(c)≤5 => we have −4≤ f(8)−f(2)/8−2 ≤5