Question

What is the maximum number of relative extrema contained in the graph of this function f(x)=3x^3-x^2+4x-2

Answer 1

the answer is 2 (apex)

Answer 2

The maximum number of turning points in a cubic function is 2.

In this case,


`f(x)=3x^3-x^2+4x-2\implies f'(x)=9x^2-2x+4`

The discriminant is
`(-2)^2-4(9)(4)=-140<0`, which means the derivative has no real roots. This means there are no critical points and thus no turning points/relative extrema.