Question

identify the inflection points and local maxima and minima of the function below and its associated graph. identify the open intervals on which the function is differentiable and is concave up and concave down.

( y=\frac{3}{11}left(x^{2}-9
ight)^{\frac{2}{3}} )

radicals as needed. use a comma to separate answers as needed.)

b. the curve is never concave up.

find the open interval(s) on which the function is differentiable and is concave down. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the curve is concave down on the open interval(s)
(simplify your answer. type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

b. the curve is never concave down.

Explanation:

Step1: Find the first - derivative

Use the chain rule \(y = a(u)^{n}\), \(y^\prime=an(u)^{n - 1}u^\prime\). Let \(u=x^{2}-9\), \(a = \frac{3}{11}\), \(n=\frac{2}{3}\). Then \(u^\prime = 2x\).

$$
LATEXBLOCK0
$$

Step2: Find the second - derivative

Use the quotient rule \(y=\frac{f(x)}{g(x)}\), \(y^\prime=\frac{f^\prime(x)g(x)-f(x)g^\prime(x)}{g^{2}(x)}\). Here \(f(x)=4x\), \(f^\prime(x) = 4\), \(g(x)=11(x^{2}-9)^{\frac{1}{3}}\), \(g^\prime(x)=11\times\frac{1}{3}(x^{2}-9)^{-\frac{2}{3}}\times2x=\frac{22x}{3}(x^{2}-9)^{-\frac{2}{3}}\)

$$
LATEXBLOCK1
$$

Step3: Determine concavity

Set \(y^{\prime\prime}<0\) (for concave down). Since \((x^{2}-9)^{\frac{4}{3}}>0\) for \(x
eq\pm3\), we solve \(x^{2}-27<0\)

$$x^{2}-27<0\Rightarrow(x - 3\sqrt{3})(x + 3\sqrt{3})<0$$

The solution of the inequality \(x^{2}-27<0\) is \(-3\sqrt{3}eq\pm3\).

Answer:

A. The curve is concave down on the open interval(s) \((-3\sqrt{3},3\sqrt{3})\)