hw 5 question 5

Question

Using the first derivative, find the interval(s) for which f(x) is increasing and decreasing.

f(x) = 12 + 9x - 3x^2 - x

Answer 1

Since f(x) = 12 + 9x - 3x^2 - x, then f'(x) = 0 + 9 - 3*2x - 1 = 8 - 6x, so

f(x) is increasing when f'(x) = 8 - 6x is > 0, so let's solve that for x:

8 > 6x, so 8/6 > x, so 4/3 > x, so x < 4/3, and

f(x) is decreasing when f'(x) = 8 - 6x is < 0, so let's solve that for x:

8 < 6x, so 8/6 < x, so 4/3 < x, so x > 4/3, so finally we have:

f(x) is increasing on (-∞, 4/3), and f(x) is decreasing on (4/3, ∞)