Question

What are the local extrema, if any, of `f (x) =a(x-2)(x-3)(x-b)`, where `a` and `b` are integers?

Answer 1

`f (x) =a(x-2)(x-3)(x-b)`

The local extrema obey

`(df)/dx = a (6 + 5 b - 2 (5 + b) x + 3 x^2) = 0`

Now, if `a ne 0` we have

`x = 1/3 (5 + b pm sqrt[7 - 5 b + b^2])`

but `7 - 5 b + b^2 gt 0` (has complex roots) so `f(x)` has allways a local minimum and a local maximum. Supposing `a ne 0`