Question

Determine the intervals on which the function is decreasing and increasing and then find local minima and maxima. f(x) = (x-2)(x+3)

Answer 1

f(x) = (x-2)(x+3)
f(x) increasing if f(x) > 0
(x-2)(x+3) > 0
x < -3 or x > 2

f(x) decreasing if f(x) < 0
(x-2)(x+3) < 0
x < 0 or x > -3

Answer 2

f(x) has a local minimum value when f'(x) = 0.
f(x) = (x-2)(x+3) = x^2 + x - 6
f'(x) = 2x + 1 = 0 when x = -1/2
f(-1/2) = -25/4
so local minimum f(x) is -25/4 when x = -1/2.