- #1
issacnewton
- 1,026
- 36
Homework Statement
Determine the values of the number ##a## for which the function ##f## has no critical number. $$f(x) = (a^2+a-6)\cos{2x} + (a-2)x + \cos{1}$$
Homework Equations
Concept of critical point
The Attempt at a Solution
Now the derivative of this function will be ##f'(x) = -2(a^2+a-6)\sin{2x} + (a-2)##. Now the amplitude of the sine function is ##2 |a^2+a-6|##. The maximum and the minimum values of the first term are ##2 |a^2+a-6|## and ##-2 |a^2+a-6|##. So ##f'(x)## will never be zero if ##|a-2| > 2 |a^2+a-6|##. And if ##f'(x)## is never zero, then the function ##f## will have no critical number. ##f'(x)## exists for all real numbers, so we don't have to worry about the points at which ##f'(x)## does not exist. So function ##f## will have no critical number if ##|a-2| > 2 |a^2+a-6|##. Solving this inequality, we get ##-\frac{7}{2} < a < -\frac{5}{2} ##. Is this correct ?