# Critical number

1. Mar 29, 2009

### ARYT

1. The problem statement, all variables and given/known data

Determine the values of the number "a" for which the function "f" has no critical number.

2. Relevant equations
If I'm not mistaken, this should be the result for differentiation:

3. The attempt at a solution

Critical points are either where function derivation is not defined (does not exists), OR where the derivation is equal to zero.
This is my answer; although, it seems to be a very complicated and somehow incorrect:

"a" should not be these values, so "f" will not have any critical number.

Last edited by a moderator: Nov 29, 2012
2. Mar 29, 2009

### HallsofIvy

Staff Emeritus
No, a is a constant not function of x!

As you say, a "critical point" is one where $f'(x)= -2(a^2+a- 6)sin(2x)+ a- 2$ does not exist or is 0. That exists for all a and is 0 when
[tex]-2(a^2+ a- 6)sin(2x)+ a- 2= 0[/itex]
so
[tex]sin(2x)= \frac{a-2}{-2(a^2+ + a- 2)}[/itex]
That never happens if
[tex]\frac{a-2}{-2(a^2+ a- 2)}> 1[/itex]