Question

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

( y = \frac { 3 } { 7 } ( x ^ { 2 } - 25 ) ^ { \frac { 2 } { 3 } } )

find the inflection points of the curve. select the correct choice below 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

Use the chain rule. If \(y = \frac{3}{7}(x^{2}-25)^{\frac{2}{3}}\), let \(u=x^{2}-25\), then \(y=\frac{3}{7}u^{\frac{2}{3}}\).
The derivative of \(y\) with respect to \(u\) is \(y_{u}'=\frac{3}{7}\times\frac{2}{3}u^{-\frac{1}{3}}=\frac{2}{7}u^{-\frac{1}{3}}\), and the derivative of \(u\) with respect to \(x\) is \(u_{x}' = 2x\).
By the chain rule \(y'=\frac{2}{7}(x^{2}-25)^{-\frac{1}{3}}\times2x=\frac{4x}{7(x^{2}-25)^{\frac{1}{3}}}\).

Step2: Find the second derivative

Use the quotient rule. If \(y'=\frac{4x}{7(x^{2}-25)^{\frac{1}{3}}}\), where \(u = 4x\), \(u'=4\) and \(v = 7(x^{2}-25)^{\frac{1}{3}}\), \(v'=\frac{7\times2x}{3(x^{2}-25)^{\frac{2}{3}}}\).
By the quotient rule \(y''=\frac{4\times7(x^{2}-25)^{\frac{1}{3}}-4x\times\frac{14x}{3(x^{2}-25)^{\frac{2}{3}}}}{49(x^{2}-25)^{\frac{2}{3}}}\).
Simplify the numerator:

$$
LATEXBLOCK0
$$

Set \(y'' = 0\), then \(x^{2}-75=0\), so \(x=\pm5\sqrt{3}\).

Step3: Check the concavity

• When \(x\in(-\infty,- 5\sqrt{3})\), let \(x=-6\sqrt{3}\), \(y''=\frac{28((-6\sqrt{3})^{2}-75)}{3((-6\sqrt{3})^{2}-25)^{\frac{2}{3}}}>0\)
• When \(x\in(-5\sqrt{3},5\sqrt{3})\), let \(x = 0\), \(y''=\frac{28(0 - 75)}{3(0 - 25)^{\frac{2}{3}}}<0\)
• When \(x\in(5\sqrt{3},\infty)\), let \(x = 6\sqrt{3}\), \(y''=\frac{28((6\sqrt{3})^{2}-75)}{3((6\sqrt{3})^{2}-25)^{\frac{2}{3}}}>0\)

Answer:

A. The inflection point(s) is/are \((-5\sqrt{3},\frac{3}{7}(75 - 25)^{\frac{2}{3}}), (5\sqrt{3},\frac{3}{7}(75 - 25)^{\frac{2}{3}})\)