Question

Assume that

f ' (x)

≤ M for a ≤ x ≤ b. One can show by the Mean Value
Theorem that

f (b) − f (a)

≤ M (b − a),
and thus

f (a) − M(b − a) ≤ f(b) ≤ f (a) + M(b − a).

Using this fact, determine a lower bound and an upper bound for

101


(Hint: Let f (x) =

x
, a = 100, and b = 101, and then find bounds for f (101) =

101
.

First, find a reasonable value for M: