monotonicity and mvt

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

A) This statement is FALSE because [-] derivatives for any function are emblematic of a decrease for that particular function.

B) This is statement is TRUE because it accurately states the Mean Value Theorem for Differentiation.

C) This statement is FALSE because it does not accurately state the Mean Value Theorem due to the differentiability for open interval (a , b) being excluded from the statement.

D) This statement is TRUE because it accurately states Rolle's Theorem for Differentiation

E) This statement is FALSE because [+] derivatives for any function are emblematic of an increase for that particular function.

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Note: There is also a Mean Value Theorem for Integrals which is distinct from the Mean Value Theorem for Differentiation.