Question

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.

( f ( x ) = ln left( 3 - 5 x ^ { 2 }
ight) )

identify the coordinates of the local maximum points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the local maximum point(s) is/are ( ( 0, ln ( 3 ) ) )
(type an ordered pair. use a comma to separate answers as needed. type an exact answer.)
b. there are no local maximum points

identify the coordinates of the local minimum points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the local minimum point(s) is/are
(type an ordered pair. use a comma to separate answers as needed. type an exact answer.)
b. there are no local minimum points

Explanation:

Step1: Find the domain

For \(y = \ln(3 - 5x^{2})\), the argument of the logarithm must be positive. So \(3-5x^{2}>0\), which gives \(x^{2}<\frac{3}{5}\) or \(x\in(-\sqrt{\frac{3}{5}},\sqrt{\frac{3}{5}})\).

Step2: Differentiate the function

Using the chain - rule, if \(y=\ln(u)\) and \(u = 3-5x^{2}\), then \(y^\prime=\frac{u^\prime}{u}\).
\(u^\prime=-10x\), so \(y^\prime=\frac{-10x}{3 - 5x^{2}}\).
Set \(y^\prime = 0\), then \(-10x=0\) gives \(x = 0\).

Step3: Second - derivative test

Differentiate \(y^\prime=\frac{-10x}{3 - 5x^{2}}\) using the quotient rule \((\frac{f}{g})^\prime=\frac{f^\prime g - fg^\prime}{g^{2}}\).
Here \(f=-10x\), \(f^\prime=-10\), \(g = 3-5x^{2}\), \(g^\prime=-10x\).
\(y^{\prime\prime}=\frac{-10(3 - 5x^{2})-(-10x)(-10x)}{(3 - 5x^{2})^{2}}=\frac{-30 + 50x^{2}-100x^{2}}{(3 - 5x^{2})^{2}}=\frac{-30 - 50x^{2}}{(3 - 5x^{2})^{2}}\).
When \(x = 0\), \(y^{\prime\prime}=\frac{-30}{9}<0\). So \(x = 0\) is a local maximum.
Since the function \(y=\ln(3 - 5x^{2})\) is a concave - down function (from the second - derivative \(y^{\prime\prime}=\frac{-30 - 50x^{2}}{(3 - 5x^{2})^{2}}<0\) for all \(x\in(-\sqrt{\frac{3}{5}},\sqrt{\frac{3}{5}})\)) and has only one critical point \(x = 0\) (from the first - derivative \(y^\prime=\frac{-10x}{3 - 5x^{2}}\)), there are no local minimum points.

Answer:

For local maximum points: A. The local maximum point(s) is/are \((0,\ln(3))\)
For local minimum points: B. There are no local minimum points