A differentiable function called f(x) achieves its maximum when x=0. Which of the following must then be true?

Question

1. The function p(x) = xf(x) has a critical point when x = 0.

2. The function m(x) = (f(x))^2 has its maximum at x = 0.

3. The function j(x) = f(x^2) has its maximum at x = 0.

Answer 1

#1. No
consider f(x) = e^(-x^2)

#2. No
consider f(x) = -x^2

See what you can do with #3.