Question

For what values of 𝑐 does the curve 𝑓(𝑥) = 2𝑥3 + 𝑐𝑥2 + 2𝑥 have maximum and minimum

points?

Answer 1

$$f'(x)=6x^2+2cx+2$$

Find the critical number(s)

$$f'(x)=0=>6x^2+2cx+2=0$$

$$3x^2+cx+1=0$$ $$x=\dfrac{-c\pm\sqrt{c^2-12}}{6}$$

We consider $x\in \R$

$$c^2-12\ge0=>c\le-\sqrt{12}\ or\ c\ge\sqrt{12}$$

The curve $𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥$ have maximum and minimum for $c\in(-\infin, -\sqrt{12})\cup (\sqrt{12}, \infin).$