Find the intervals on which f(x) is increasing and the intervals on which f(x) is decreasing.

Question

The problem is F(x) = -x^2+8 ln x

Answer 1

Shawn,

The domain of f(x) is (0, infinity). Then we need to find the critical values on this interval first.

f'(x) = 2x + 8/x. The first derivative does not exist at x=0, but it is not in the domain.

Set f'(x) = 0

2x+8/x = 0

2x = - 8/x

2x² = - 8

x² = - 4

This has no solution -- > there is no critical number, Check the sign of the first derivative at any positive number (the domain of f(x), and you find f'(x) > 0.

Since there could be no sign change of f', the function is increasing on its entire domain.