Question

A function is defined by the polynomial 𝑓(𝑥) = 3𝑥⁴ − 4𝑥³ − 12𝑥² + 8. Find and classify all the stationary points of f(x).

Answer 1

Let us find and classify all the stationary points of the function 𝑓(𝑥) = 3𝑥⁴ − 4𝑥³ − 12𝑥² + 8. The function is differetiable in all points of the real line. Let us find the points $x$ for which $f'(x)=0.$ Since $f'(x)=12x^3-12x^2-24x,$ we conclude that $12x^3-12x^2-24x=0$ implies $12x(x^2-x-2)=0$, and hence $12x(x+1)(x-2)=0$. It follows that $x_1=-1, \ x_2=0,\ x_3=2$ are the stationary points of the function 𝑓(𝑥) = 3𝑥⁴ − 4𝑥³ − 12𝑥² + 8. Taking into account that the function $f$ is continuous and $f'(-2)=-96<0,\ f'(-0.5)=7.5>0,$

$f'(1)=-24<0,\ f'(3)=144>0,$ we conclude that the function is increasing on the intervals $(-1,0)$ and $(2,+\infty),$ and the function is deccreasing on the intervals $(-\infty,-1)$ and $(0,2).$

Therefore, $x_1=-1,\ x_3=2$ are the points of minimum, and $x_2=0$ is the point of maximum.