Answer to Question #5533 in Calculus for natalie

Question

for y=x^4-72x^2-17 use analytic methods to find the exact intervals on which the function is increasing decreasing concave up and down and find the extreme values and inflection points

Answer 1

Let's find the extreme values and inflection points
<img style="width: 472px; height: 421px;" 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" 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