mean value theorem

Question

Select the true statements involving the mean value theorem and monotonicity.

a) If 𝑓′(𝑥)<0 for all 𝑥 in an interval, then 𝑓 is increasing on that interval.

b) If 𝑓 is continuous on [𝑎,𝑏] and differentiable on (𝑎,𝑏), then there exists at least one 𝑐 in (𝑎,𝑏) such that 𝑓(𝑏)−𝑓(𝑎)=𝑓′(𝑐)(𝑏−𝑎).

c) If 𝑓 is continuous on [𝑎,𝑏], then there exists at least one 𝑐 in (𝑎,𝑏) such that 𝑓′(𝑐)=𝑓(𝑏)−𝑓(𝑎)/𝑏−𝑎.

d )If 𝑓 is differentiable and 𝑓′(𝑥)=0 for all 𝑥∈(𝑎,𝑏), then 𝑓 is constant on [𝑎,𝑏].

e) If 𝑓′(𝑥)>0 for all 𝑥 in an interval, then 𝑓 is decreasing on that interval

Answer 1