# Critical points

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Let g(x) = 300-8x^3-x^4

-Find the local maximum and minimum values.

-Find the intervals of concavity and the inflection points.

-Use this information to carefully sketch the graph of g.

https://brainmass.com/math/calculus-and-analysis/calculating-critical-points-sketching-function-2918

#### Solution Preview

g'(x)= -24x^2 - 4x^3 = -4x^2 (6 + x)

g'(x)=0 => x=0 or x=-6 therefore (0,300) and (-6,732) are the local extrema.

g''(x)= -48x - 12x^2 = -12x (4 + x)

g''(x)=0 => x=0 or x=-4, therefore (0,300) and (-4,556) are inflection points.

g''(1)= -60, g''(-1)= 36, g''(-6)=-144,

Therefore (0,oo) is ...

#### Solution Summary

This shows how to find points of inflection, local maximum and minimum, and use these to sketch the graph.

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