extrema critical numbers

Question

Which of the given statements involving extrema are true?

a. If 𝑐 is a critical number, then there is a local extreme value at 𝑐.

b. If a function 𝑓 is defined on the interval [𝑎,𝑏], differentiable on (𝑎,𝑏), and decreasing on [𝑎,𝑏], then there must be an absolute minimum at 𝑏.

c.If a function 𝑓 fails to be differentiable at 𝑐, then it cannot have a local extreme value at 𝑐.

d.Any function 𝑓 that is defined on an open interval (𝑎,𝑏) must have an absolute maximum. 

e.Any function 𝑓 that is defined and continuous on a closed interval [𝑎,𝑏] must have an absolute minimum.

Answer 1

For true/false questions, you can build intuition by trying to come up with a counterexample or a proof for each statement. Answers and explanations for each part follow.

a) False. For a counterexample, consider the function x³ --- it has a critical point at x=0, where its derivative is zero, but no local extrema. (This will always happen when a critical point coincides with an inflection point).

b) True. Since f(x) is decreasing on [a,b], we have f(x) < f(b) for any x < b in [a, b), making b an absolute minimum.

c) False. The function x^2/3 has a critical point at x=0, which is also a local minimum (and also an absolute minimum).

d) False. The function x^-1 is defined on the interval (0, 1), but has no absolute maximum, because of the vertical asymptote

e) True. This is the Extreme Value Theorem. A college course in Analysis usually covers the full proof of the theorem. The intuitive reasoning behind the theorem is as follows:

1. a continuous function sends closed, bounded intervals in its domain to closed, bounded intervals in its range
2. such an interval in the range has a minimum value
3. there must be a point in the domain corresponding to the aforementioned minimum value

Answer 2

a is false (example: f(x) = x³ has a critical value at x = 0 but no local extreme value at x = 0).

c is false (example: f(x) = lxl is not differentiable at x = 0 but has a minimum there)

d is false (example: f(x) = tanx is defined on (-π/2, π/2) but doesn't have a max on that interval)