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

Nov 22, 2016

$f \left(x\right) = a \left(x - 2\right) \left(x - 3\right) \left(x - b\right)$

The local extrema obey

$\frac{\mathrm{df}}{\mathrm{dx}} = a \left(6 + 5 b - 2 \left(5 + b\right) x + 3 {x}^{2}\right) = 0$

Now, if $a \ne 0$ we have

$x = \frac{1}{3} \left(5 + b \pm \sqrt{7 - 5 b + {b}^{2}}\right)$

but $7 - 5 b + {b}^{2} > 0$ (has complex roots) so $f \left(x\right)$ has allways a local minimum and a local maximum. Supposing $a \ne 0$