Find the intervals on which f is increasing and decreasing.

Question

Find the intervals on which f is increasing and decreasing.

f(x) = -3x^2 + 54 ln x

Answer 1

Domain of this function x > 0. Then derivative f'(x) = - 6x + 54/x = (- 6x² + 54)/x.

Now, f(x) increasing if f'(x) > 0 or (- 6x² + 54)/x > 0, because x > 0 - 6x² + 54 > 0, x² < 9, 0 < x < 3.

So interval (0,3) is interval of increasing.

f( x) decreasing if f'(x) < 0, so (- 6x² + 54)/x < 0. Becouse x > 0 - 6x² + 54 < 0, x² > 9, x > 3. So, interval (3, ∞)

is interval of decreasing

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