1. ## local extrema

Hi, I don't know how to find the local extrema, and I'm not sure if I differentiated correctly. for the first one is it -ln [(1-x)^2]-2? how do I find the critical point by letting it 0 =S would somebody be able to show me full solutions for part and b please?

2. part b) $f(x) = cos(2x)+ sin(x)$

$f'(x) = cos(x)- 2sin(2x)$

for extreme value

$0 = cos(x)-2sin(2x)$

$0 = cos(x)-2\times 2sin(x)cos(x)$

$0 = cos(x)-4sin(x)cos(x)$

$0 = cos(x)(1-4sin(x))$

$0 = 1-4sin(x)$

$1 = 4sin(x)$

$sin(x)=\frac{1}{4}$

can you solve from here?
The text passage contains a student asking for help with finding local extrema and differentiating a function, and then provides a detailed solution for part b) of the problem, which involves finding the critical points of a trigonometric function.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. ## local extrema

Hi, I don't know how to find the local extrema, and I'm not sure if I differentiated correctly. for the first one is it -ln [(1-x)^2]-2? how do I find the critical point by letting it 0 =S would somebody be able to show me full solutions for part and b please?

2. part b) $f(x) = cos(2x)+ sin(x)$

$f'(x) = cos(x)- 2sin(2x)$

for extreme value

$0 = cos(x)-2sin(2x)$

$0 = cos(x)-2\times 2sin(x)cos(x)$

$0 = cos(x)-4sin(x)cos(x)$

$0 = cos(x)(1-4sin(x))$

$0 = 1-4sin(x)$

$1 = 4sin(x)$

$sin(x)=\frac{1}{4}$

can you solve from here?
user
The text passage contains a student asking for help with finding local extrema and differentiating a function, and then provides a detailed solution for part b) of the problem, which involves finding the critical points of a trigonometric function.

TYPE: SINGLE_QUESTION_WITH_ANSWER