Using the MVT prove that if f is strictly on a b then fc Sol

Using the MVT, prove that if f is strictly on (a, b), then f\'(c)

Solution

f(a)>f(b) , since f is strictly decreasing

If f is not differentiable then f\'(c) is not defined for all c in (a,b)

If f is differentiable then by Mean Value Theorem

there is some c in (a,b) so that

f\'(c)=(f(b)-f(a))/(b-a)<0

Hence proved