Derivative of f(x) to find its maximum and minimum values

[DryRun]

Homework Statement
If λ is a positive constant, determine the maximum and minimum values of
f(x) = 9(4-3x^2)(λ-λ^-1-x)
and show that the difference between them is 4(λ+λ^-1)^3. Find the least value of this difference as the parameter λ is varied.

The attempt at a solution
I expanded the right-hand side and then did a first d.w.r.t.x

dy/dx = 81x^2 + 54x/λ - 54λx -36

I have to equate this to zero, to find either the minimum or maximum value:

81x^2 + 54x/λ - 54λx -36 = 0

But i don't know how to proceed next, since the value of λ is unknown, so i cannot solve for x.

 

[HallsofIvy]

You can certainly use the quadratic formula to solve for x. Of course those values, and the minimum and maximum values of the function, will depend upon \lamba.

 

[Mark44]

Mod note - moved from Precalc section.