determine the value of a for which the function f has no critical number:

f(x)= (a^{2}+a-6)cos2x+(a-2)x+cos1

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- Nov 6th 2012, 10:09 AMpnfullercritical number?
determine the value of a for which the function f has no critical number:

f(x)= (a^{2}+a-6)cos2x+(a-2)x+cos1 - Nov 6th 2012, 11:29 AMMarkFLRe: critical number?
What do you find when you equate the derivative to zero? Is the last term actually cos(1)?

- Nov 6th 2012, 11:48 AMpnfullerRe: critical number?
yes the last term is actually cos(1) and how do you take the derivative and set it equal to zero? is a just a constant?

- Nov 6th 2012, 11:51 AMMarkFLRe: critical number?
Yes, you would treat $\displaystyle a$ as a constant.

- Nov 6th 2012, 12:09 PMpnfullerRe: critical number?
i got the derivative being

-2a^2sin2x-2asin2x+12sin2x+a-2 and then i dont know how to take the derivative of cos1 and i dont know how to set this derivative equal to 0 with the constants in it? - Nov 6th 2012, 12:18 PMMarkFLRe: critical number?
cos(1) is a constant, so its derivative is zero. I would write the derivative as:

$\displaystyle f'(x)=-2(a^2+a-6)\sin(2x)+(a-2)=-2(a+3)(a-2)\sin(2x)+(a-2)=$

$\displaystyle (a-2)(-2(a+3)\sin(2x)+1)=0$

What condition on $\displaystyle a$ causes the second factor to have no root? - Nov 6th 2012, 12:21 PMpnfullerRe: critical number?
- Nov 6th 2012, 12:24 PMMarkFLRe: critical number?
I accidentally hit Post Quick Reply rather than Go Advanced...I have edited my post.

- Nov 6th 2012, 12:36 PMpnfullerRe: critical number?
i dont know what you mean about the root and i still dont understand how to set it equal to zero and solve for whatever like that?

- Nov 6th 2012, 01:04 PMMarkFLRe: critical number?
A root is a value that causes an expression to be zero. Anyway, consider:

$\displaystyle -1\le\sin(2x)\le1$

Now, equate the second factor to zero, solve for $\displaystyle \sin(2x)$ and substitute into the above inequality. This will tell the the values of $\displaystyle a$ for which there are critical numbers. - Nov 6th 2012, 03:05 PMpnfullerRe: critical number?
so i got a=2 and then the second part (a+3)sin(2x)=1/2 and i dont know how to simplify it any further

- Nov 6th 2012, 03:18 PMMarkFLRe: critical number?
When

*a=2*, your function is constant, so there will be no critical values.

As for the second part, solve for sin(2x), then as an alternate to what I suggested before, set the magnitude of this to greater than 1. You will have 2 cases to consider:

i) $\displaystyle \frac{1}{2(a+3)}>1$

ii) $\displaystyle \frac{1}{2(a+3)}<-1$ - Nov 6th 2012, 03:22 PMpnfullerRe: critical number?
i dont understand

- Nov 6th 2012, 03:30 PMMarkFLRe: critical number?
We found:

$\displaystyle \sin(2x)=\frac{1}{2(a+3)}$

We therefore know that if $\displaystyle \left|\frac{1}{2(a+3)} \right|>1$ then there is no solution, that is, there is no real value of $\displaystyle a$ which will give us a critical value. - Nov 6th 2012, 03:39 PMpnfullerRe: critical number?
well how do you solve for a?