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.
I started by taking the derivative first, since the derivative becomes 0 at any local maxima/minima.
$$f'(x)=(a-2)+2(a-1)x-3x^2=0$$
I tried to get the roots using quadratic formula which just gives an expression in $a$ which doesn't seem to conclude at any desired result. Also, I don't understand how I can separately apply this condition for $x>1$ and $x<1$ respectively.
Rolle's Mean Value Theorem also struck me since I could use it to conclude that $f'(x)$ has a root but I would need an interval for that,
to get the condition $f(p)=f(q)$ so that $f'(x)$ has a root between $p$ and $q$. Clearly, $-1$ and $+1$ would be one of these in each case but what about the other end of the interval? I'm not able to find any other x which gives the same value as $f(-1)$ or $f(+1)$
Please help me out with a way to approach this, or if there's any other better way.