## How many integer values ​​of parameter (m) are there for the function (y = – {x^4} + 6{x^2} + mx) to have three extremes? – Math book

How many integer values ​​of the parameter $$m$$ so that the function $$y = – {x^4} + 6{x^2} + mx$$ have three extreme points?

A. $$17$$.

B. $$15$$.

C. $$3$$.

D. $$7$$.

Select REMOVE

We have: $$y’ = – 4{x^3} + 12x + m$$. Consider the equation $$y’ = 0 \Leftrightarrow – 4{x^3} + 12x + m = 0\,\,\,\,\,\,\left( 1 \right)$$.

For the function to have three extreme points, the equation $$\left( 1 \right)$$ must have 3 distinct solutions.
We have: $$\left( 1 \right) \Leftrightarrow m = 4{x^3} – 12x$$.
Consider the function $$g\left( x \right) = 4{x^3} – 12x$$ Have $$g’\left( x \right) = 12{x^2} – 12$$. Give $$g’\left( x \right) = 0 \Leftrightarrow 12{x^2} – 12 = 0 \Leftrightarrow x = \pm 1$$.
Variation table of $$g\left( x \right)$$