How do you find intercepts, extrema, points of inflections, asymptotes and graph $g(x)=x+32/x^2$ ?

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Answer 1 · 2017-12-01T17:23:57

Please see below for a partial solution.

$g(x) = x+32/x^2$

x intercepts
$g(x) = 0$ at $x=-32/x^2$
which happens at $x^3 = -32$

so $x = root(3)(-32) = -2root(3)4$

y intercept
None. $g(0)$ does not exist.

Asymptotes

$lim_(xrarr0)g(x) = oo$ , so $x=0$ (the $y$ -axis) is a verticle asymptote..

$lim_(xrarr00)g(x) = oo$ so there is no horizontal asymptote.

$lim_(xrarroo)(g(x)-x) = 0$ so $y=x$ is an oblique (slant) asymptote)

Analysis of first derivative

$g'(x) = 1-64/x^3 = (x^3-64)/x^3$ is undefined at $x=0$ and is $0$ at $x=4$ .

On $(-oo,0)$ , we have $g'(x) > 0$ so $g$ is increasing.
$x=0$ is not a critical number.

On $(0,4)$ , we have $g'(x) < 0$ so $g$ is decreasing.
On $(4,oo)$ , we have $g'(x) > 0$ so $g$ is increasing.

$f(4)=6$ is a local minimum.

How do you find intercepts, extrema, points of inflections, asymptotes and graph g(x)=x+32/x^2g(x)=x+32/x^2?