Question

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

$f(x)=\ln (3 - 5x^{2})$

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

a. the absolute 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 absolute minimum points

find the inflection points. 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. use a comma to separate answers as needed. type an
exact answer.)
b. there are no inflection 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: Find the first - derivative

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

Step3: Find the second - derivative

Using the quotient rule \((\frac{f}{g})^\prime=\frac{f^\prime g - fg^\prime}{g^{2}}\), where \(f=-10x\) and \(g = 3-5x^{2}\). Then \(f^\prime=-10\) and \(g^\prime=-10x\).

$$
LATEXBLOCK0
$$

Step4: Analyze the second - derivative for inflection points

Set \(y^{\prime\prime}=0\). The numerator \(-10(3 + 5x^{2}) = 0\) has no real solutions because \(3+5x^{2}>0\) for all real \(x\).

Answer:

B. There are no inflection points.