# `f(x) = (x^2) - x - ln(x)` (a) Find the intervals on which `f` is increasing or decreasing. (b) Find the local maximum and minimum values of `f`. (c) Find the intervals of concavity and the...

`f(x) = (x^2) - x - ln(x)` (a) Find the intervals on which `f` is increasing or decreasing. (b) Find the local maximum and minimum values of `f`. (c) Find the intervals of concavity and the inflection points.

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Expert Answers

Borys Shumyatskiy | Certified Educator

The function is defined for x>0 and is infinitely differentiable.

`f'(x) = 2x - 1 - 1/x.`

f'(x) = 0 is equivalent to `2x^2 - x - 1 = 0,` or x=1, x=-1/2. The second x is out of domain.

(a) for x in (0, 1) f'(x) < 0 and f is decreasing, for x in `(1, +oo)` f'(x)>0 and f is increasing.

(b) therefore x=1 is a local minimum, f(1) = 0.

(c) `f''(x) = 2 + 1/x^2` which is >0. So f is concave upward.

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