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

Please see below for a partial solution.

Asymptotes

Analysis of first derivative

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To find the intercepts:

- x-intercept: Set ( g(x) = 0 ) and solve for ( x ).
- y-intercept: Evaluate ( g(x) ) when ( x = 0 ).

To find extrema:

- Take the derivative of ( g(x) ), ( g'(x) ).
- Set ( g'(x) = 0 ) and solve for ( x ) to find critical points.
- Test these critical points to determine if they correspond to local extrema.

To find points of inflection:

- Take the second derivative of ( g(x) ), ( g''(x) ).
- Set ( g''(x) = 0 ) and solve for ( x ) to find possible inflection points.
- Test these points to confirm if they are points of inflection.

To find asymptotes:

- Horizontal asymptote: As ( x ) approaches positive or negative infinity, find the limit of ( g(x) ).
- Vertical asymptotes: Find values of ( x ) where the denominator of ( g(x) ) equals zero.

Graph ( g(x) = x + \frac{32}{x^2} ) using the information gathered from intercepts, extrema, points of inflection, and asymptotes.

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