Relative minimum, points of inflection, and absolute maximum
The derivative of a function f is given by f'(x) = (x^3-2x)(cos x) for 0 ≤ x ≤ 2.
a. Find the x-coordinate of the relative minimum of f(x). You may use your calculator, but show the analysis that leads to your conclusion.
b. Find the x-coordinate of each point of inflection on the graph f(x). Justify your answer.
c. Find the x-coordinate of the point at which f(x) attains an absolute maximum. Justify your answer.
a. I began by solving f'(x) = 0. I found x = 0, 2^(1/2), π/2. I used test points to find that π/2 is the relative minimum. Is this correct?
b. I got the second derivative: f"(x)=(3x^2-2)cos(x)-x(x^2-2)sin(x)
Then, I solved for f"(x)=0. I got x=1.49568 and .64224. Is this correct?
c. I'm not sure how to go about this one. Since I used test points before in part a, I know which are the relative extrema, but how do I know which is the absolute maximum?