Question

#17 - Step by Step??

Answer 1

`f(x)=(x-1)(x-4)^2`


`f'(x)=(x-4)^2+2(x-1)(x-4)=(x-4)(x-4+2x-2)=(x-4)(3x-6)=3(x-4)(x-2)`

The first derivative is zero when
`x=2` and
`x=4`, so you need to only check the sign of the derivative in the intervals
`(-\infty,2)`,
`(2,4)`, and
`(4,\infty)`.

Since
`f'(x)` describes a convex function (a parabola that opens upward), and you know the location of the roots, you know that
`f'(x)<0` on the interval
`(2,4)`, and so the function is decreasing on this interval only.