Let f (x) = 2x3 + 9x2 − 24x − 10. Find the intervals on which f (x) is increasing or decreasing.

Answer 1

f(x) = 2x³ + 9x² - 24x -10 given

f ' (x) = 6x² + 18x - 24 first derivative

f '' (x) = 12x + 18 second derivative

first derivative test for local maxima

f ' (x) = 6x² + 18x - 24 = 0 set first derivative to equal zero

x² + 3x - 4 = 0 divide all terms by 6

(x + 4) (x - 1) = 0 factor

x = -4 and x = -1 maxima at x = - 4 and x = -1

second derivative test for concavity at maxima

f '' (x) = 12x + 18 second derivative

f '' (-4) = 12(-4) + 18 = -48 + 18 = -30 negative answer: concave down: maximum point at x = -4

f '' (1) = 12(1) + 18 = 12 + 18 = 30 positive answer: concave up: minimum point at x = 1

therefore,

increasing (-infinity, -4);

decreasing (-4, 1);

increasing (1, infinity)