Critical number, max & minimum value

In summary, the conversation was about finding the absolute minimum and maximum values of a function and finding the critical numbers for another function. The attempt at a solution involved using the derivative and solving for x, but there were some algebra errors that needed to be corrected. The solution for the first problem did not take into account the double angle formula for cosine, and the solution for the second problem had an incorrect numerator. The correct critical numbers for the second problem are x=2 and x=3.
  • #1
catjay
3
0
Hey guys, I seem to have some troubles solving these problems. I was hoping if you guys can give me some insight. Thank you so much for your help!

Homework Statement


Find the absolute min and max value of each function


Homework Equations


f(x) = sin 2x - 2 sin x on (-pi,pi)


The Attempt at a Solution


finding the critical #
f'(x) = 2 cos 2x - 2 cos x
= 2 cos x
set the equation = 0
0 = 2 cos x
x = +pi/2, -pi/2



However, this is not the right answer because when I graphed it, the absolute minimum value was 2pi/3 while the absolute max was squaroot 3/2. I was wondering which step I did wrong? Thank!

I also have another question, in which I'm not sure why I got the answer wrong. The critical # supposed to be x=2,3 while I got x=3,6.

Homework Statement


Find the critical #

Homework Equations



f(x) = x (3-x)^(1/2)

The Attempt at a Solution



f'(x)=[x/(2 * squareroot (3-x)] + [squareroot (3-x)]
finding common denominator:
f'(x) = [x + 2(3-x)]/[2 * squareroot (3-x)]
=(6-x)/[2 * squareroot (3-x)]

Therefore, the critical #s are:
6-x=0 -----> x=6
2 * squareroot (3-x) = 0 ------> x=3
 
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  • #2
For your first problem, 2 cos 2x - 2 cos x ≠ 2 cos x.

For your second I plugged your function into a derivative calculator and got a different numerator, so you made an algebra error there as well.
 
  • #3
As for the first one, is there any identity I'm not aware of?

As for the second question, the numerator was supposed to be 6-3x. I tried to conjugate it but was unable to get that number.

:/
 
  • #4
Well you could use the double angle formula on the cos2x which may simplify it some.. you want to set f'(x) equal to zero and solve for x, so maybe play around with that identity or with it as is and see if you can deduce what values of x satisfy the equation.

As for the second I'm not sure why you need to conjugate, you should apply the product rule for derivatives and then simplify by giving all the terms a common denominator.
 
  • #5
I think I know what I did wrong. Thanks!
 

1. What is a critical number?

A critical number is a point on a function where the derivative is equal to zero or does not exist. It is used to find the maximum or minimum values of a function.

2. How is a critical number related to a maximum or minimum value?

At a critical number, the derivative of a function is equal to zero or undefined. This means that the function is either increasing or decreasing at that point, which can help determine whether it is a maximum or minimum value.

3. What is the difference between a maximum value and a minimum value?

A maximum value is the highest point on a function, while a minimum value is the lowest point. These values are important in finding the optimal solutions in various applications, such as optimization problems in mathematics and economics.

4. How can critical numbers be used to find the maximum or minimum values of a function?

By setting the derivative of a function equal to zero and solving for the critical number(s), we can determine the x-values where the function may have a maximum or minimum value. Then, by plugging these values into the original function, we can find the corresponding y-values.

5. Can a function have multiple critical numbers?

Yes, a function can have multiple critical numbers. It is important to check each critical number to determine whether it is a maximum or minimum value. In some cases, there may also be no critical numbers, meaning the function does not have a maximum or minimum value.

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