# Critical Numbers

• Oct 30th 2008, 01:04 PM
dm10
Critical Numbers
I need some help with these problems I have to find the critical numbers fo the problems.

First I need to differentiate the problem and then I need to set it equal to 0 and solve for the variable.

The first one I just need help differentiating:

f(theta)=2cos(theta)+sin^2(theta)

This second one I differentiated but I don't know how to solve for theta:

g(theta)=4(theta)-tan(theta)
and
g'(theta)=4-sec^2(theta)

I appreciate any help.
• Oct 30th 2008, 01:39 PM
ajj86
Solution
Well, for the first one, you take the derivative of each term
Keep in mind:
The derivative of cosx = -sinx
The derivative of sinx = cosx

The first term's derivative would be 2 * -sin(theta) = -2sin(theta)
The second term's derivative involves the chain rule
It is gotten by first using the power rule, and then differentiating the trig function inside the parenthesis:
So, it is 2sin(theta)*cos(theta)

So, f'(theta) = -2sin(theta) +2sin(theta)cos(theta)

I'll leave the factoring and solving for theta to you.
• Oct 30th 2008, 01:43 PM
ajj86
Second Question
g'(theta)=4-sec^2(theta)

So, you set the right side equal to zero:
4 - sec^2(theta) = 0

Using a little algebra:
sec^2(theta) = 4

Take the square root of both sides to get:
sec(theta) = 2

Now remember, sec(theta) = 1/cos(theta)
So, 1/cos(theta) = 2

Take the inverse of both sides to get:
cos(theta) = 1/2

I'm sure you know how to do the rest. I hope this helps.
Also, for the first problem, do some review of the chain rule. It will help you understand how to get the derivatives a lot better.
• Oct 30th 2008, 06:16 PM
dm10
Quote:

Originally Posted by ajj86
g'(theta)=4-sec^2(theta)

So, you set the right side equal to zero:
4 - sec^2(theta) = 0

Using a little algebra:
sec^2(theta) = 4

Take the square root of both sides to get:
sec(theta) = 2

Now remember, sec(theta) = 1/cos(theta)
So, 1/cos(theta) = 2

Take the inverse of both sides to get:
cos(theta) = 1/2

I'm sure you know how to do the rest. I hope this helps.
Also, for the first problem, do some review of the chain rule. It will help you understand how to get the derivatives a lot better.

Acutally I don't know what to do next. That's where I'm stuck.
• Nov 2nd 2008, 05:38 PM
ajj86