# What are the local extrema of f(x) = tan(x)/x^2+2x^3-x?

Jan 16, 2017

Near $\pm 1.7$. See graph that gives this approximation. I would try to give more precise values, later.

#### Explanation:

The first graph reveals the asymptotes $x = 0 , \pm \frac{\pi}{2} \pm \frac{3}{2} \pi , \pm \frac{5}{2} \pi , . .$

Note that $\tan \frac{x}{x} ^ 2 = \left(\frac{1}{x}\right) \left(\tan \frac{x}{x}\right)$

has the limit $\pm \infty$, as $x \to {0}_{\pm}$

The second (not-to-scale ad hoc ) graph approximates local extrema

as $\pm 1.7$. I would improve these, later.

There are no global extrema.
graph{tan x/x^2+2x^3-x [-20, 20, -10, 10]}
graph{tan x/x^2+2x^3-x [-2, 2, -5, 5]}