Question

identify the inflection points and local maxima and minima of the graphed function. identify the open intervals on which the function is differentiable and is concave up and concave down.
find the inflection point(s). select the correct choice and, if necessary, fill in the answer box to complete your choice
a. the inflection point(s) is/are
(type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed )
b. there are no inflection points

Explanation:

Step1: Find the first derivative

Given \(y = \frac{x^{4}}{4}-2x^{2}-5\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(y^\prime=\frac{4x^{3}}{4}-4x=x^{3}-4x\).

Step2: Find the second derivative

Differentiate \(y^\prime=x^{3}-4x\) with respect to \(x\).
Using the power rule again, \(y^{\prime\prime}=3x^{2}-4\).

Step3: Find the inflection points

Set \(y^{\prime\prime}=0\), so \(3x^{2}-4 = 0\).
Solve for \(x\):

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When \(x = \frac{2\sqrt{3}}{3}\), \(y=\frac{(\frac{2\sqrt{3}}{3})^{4}}{4}-2(\frac{2\sqrt{3}}{3})^{2}-5=\frac{\frac{16\times9}{81}}{4}-2\times\frac{4\times3}{9}-5=\frac{4}{9}-\frac{8}{3}-5=\frac{4 - 24 - 45}{9}=-\frac{65}{9}\)
When \(x=-\frac{2\sqrt{3}}{3}\), \(y =-\frac{65}{9}\) (since the function \(y = \frac{x^{4}}{4}-2x^{2}-5\) is even, \(y(-x)=y(x)\))

Answer:

A. The inflection point(s) is/are \((-\frac{2\sqrt{3}}{3},-\frac{65}{9}),(\frac{2\sqrt{3}}{3},-\frac{65}{9})\)