Find the set of all the possible values of $a$ for which the function $f(x) = 5 + (a-2)x + (a-1)x^2 - x^3$ has a local minimum value at some $x < 1$ and local maximum value at some $x > 1$
The first derivative of f(x) is :
$f'(x) = (a-2) + 2x(a-1) -3x^2$
I do know the first derivative test for local maxima and local minima, but I can't figure out how I could use monotonicity to find intervals of increase and decrease of $f'(x)$
The expression for $f'(x)$ might suggest the double derivative test is the key, considering $f''(x) = 2(a-1) - 6x$ for which the intervals where it is greater than zero and less than zero can be easily found, but then again I can't think of a way how I could find a $c$ such that $f'(c) = 0$.