f(x) =x^3−3x+ 2. 1)What are the critical points of f? On what intervals is f increasing or decreasing? At what values ofx, if any, does f assume local maximum and minimum values?

Answer 1

f'(x) = 3x² - 3 = 0; x² = 1; x = - 1 or x = 1 are 2 critical numbers;

f'(x) = 3x² - 3 > 0; x < - 1 or x > 1 f(x) increasing;

f'(x) = 3x² - 3 < 0; - 1 < x < 1 f(x) decreasing

f''(x) = 6x; f''(- 1) = - 6 < 0 so, we have at x = - 1 local max = f(- 1) = - 1 + 3 + 2 = 4; (-1, 4) is local max

f''(1) = 6 > 0, so at x = 1 f(x) has min f(1) = 1 - 3 + 2 = 0; (1, 0) is local min