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 } ( x ^ { 2 } - 9 ) ^ { \frac { 2 } { 3 } } )

separate answers as needed.)

b. there are no local maxima.

find the local minima. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the local minimum/minima occur(s) at
(type an ordered pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

b. there are no local minima.

Explanation:

Step1: Find the first derivative

Use the chain rule \(y = \frac{3}{11}(x^{2}-9)^{\frac{2}{3}}\). Let \(u=x^{2}-9\), then \(y=\frac{3}{11}u^{\frac{2}{3}}\).
The derivative of \(y\) with respect to \(u\) is \(y_{u}=\frac{3}{11}\times\frac{2}{3}u^{-\frac{1}{3}}=\frac{2}{11}u^{-\frac{1}{3}}\), and the derivative of \(u\) with respect to \(x\) is \(u_{x} = 2x\).
By the chain rule \(y^{\prime}=\frac{2}{11}(x^{2}-9)^{-\frac{1}{3}}\times2x=\frac{4x}{11(x^{2}-9)^{\frac{1}{3}}}\).
Set \(y^{\prime}=0\), then \(4x = 0\) gives \(x = 0\). The derivative is undefined when \(x^{2}-9=0\), i.e., \(x=\pm3\).

Step2: Analyze the sign of the first derivative

• For \(x<- 3\), let \(x=-4\), then \(y^{\prime}=\frac{4\times(-4)}{11((-4)^{2}-9)^{\frac{1}{3}}}=\frac{-16}{11(7)^{\frac{1}{3}}}<0\).
• For \(-3
• For \(0
• For \(x>3\), let \(x = 4\), then \(y^{\prime}=\frac{4\times4}{11((4)^{2}-9)^{\frac{1}{3}}}=\frac{16}{11(7)^{\frac{1}{3}}}>0\).

Since the function changes from decreasing to increasing at \(x=-3\) and \(x = 3\).

Step3: Calculate the function values at critical points

When \(x=-3\), \(y=\frac{3}{11}((-3)^{2}-9)^{\frac{2}{3}}=0\).
When \(x = 3\), \(y=\frac{3}{11}(3^{2}-9)^{\frac{2}{3}}=0\).

Answer:

A. The local minimum/minima occur(s) at \((-3,0),(3,0)\)