The derivative of the function g(x) = x^5 - x^4 reaches its minimum value when x =

Question

Answer choices:

A) 0

B) 3/5

C) 4

D) There is no minimum.

Could someone explain how to solve this? My friend and I are very confused.

Answer 1

g(x) = x⁵ - x⁴

g'(x) = 5x⁴ - 4x³

g"(x) = 20x³ - 12x² = 4x²(5x - 3) = 0 when x = 0 or 3/5

When x < 0, g"(x) < 0. So, g'(x) is decreasing.

When 0 < x < 3/5, g"(x) < 0. So, g'(x) is decreasing.

When x > 3/5, g"(x) > 0. So, g'(x) is increasing.

g'(x) is minimized when x = 3/5.