Thread: Relative minimum, points of inflection, and absolute maximum

1. 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?

2. Originally Posted by NandP
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.

a relative max occurs where f'(x) changes sign from positive to negative

b. Find the x-coordinate of each point of inflection on the graph f(x). Justify your answer.

an inflection point occurs on the graph of f(x) wherever f'(x) changes the sign of its slope

c. Find the x-coordinate of the point at which f(x) attains an absolute maximum. Justify your answer.

absolute extrema may occur at relative extrema, or at an endpoint.

note that the graph of f'(x) is mostly negative

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