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 each local maximum. select the correct choice and, if necessary, fill in the answer box to complete your choice
a. there is one local maximum at the point
(type an ordered pair. type an exact answer, using radicals as needed)
b. there are two local maxima. the points are
(type ordered pairs. type exact answers, using radicals as needed. use a comma to separate answers.)
c. there are no local maxima.

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 \(y^\prime=x^{3}-4x=x(x^{2}-4)=x(x - 2)(x + 2)\).
Set \(y^\prime = 0\), then \(x=-2,0,2\).

Step2: Use the second - derivative test

Find the second derivative \(y^{\prime\prime}=3x^{2}-4\).
When \(x=-2\), \(y^{\prime\prime}=3\times(-2)^{2}-4=8>0\), so \(x = - 2\) is a local minimum.
When \(x = 0\), \(y^{\prime\prime}=3\times0^{2}-4=-4<0\).
When \(x = 2\), \(y^{\prime\prime}=3\times2^{2}-4=8>0\), so \(x = 2\) is a local minimum.

Answer:

C. There are no local maxima.