Question

A function is defined by the polynomial 𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 − 12𝑥 2 + 8. Find and classify all the stationary points of f(x)

Answer 1

$$f(x)=3x^4-4x^3-12x^2+8$$

Domain: $(-\infin, \infin)$

$$f'(x)=(3x^4-4x^3-12x^2+8)'$$

$$=12x^3-12x^2-24x$$

$$=12x(x^2-x-2)$$

Find the critical number(s)

$$f'(x)=0=>12x(x^2-x-2)=0$$

$$12x(x+1)(x-2)=0$$

Critical numbers: $-1,0,2.$

If $x<-1, f'(x)<0, f(x)$ decreases.

If $-1<x<0, f'(x)>0, f(x)$ increases.

If $0<x<2, f'(x)<0, f(x)$ decreases.

If $x>2, f'(x)>0, f(x)$ increases.

$f(-1)=3$

$f(0)=8$

$f(2)=-24$

The function $f(x)$ has a local maximum with value of $8$ at $x=0.$

The function $f(x)$ has a local minimum with value of $3$ at $x=-1.$

The function $f(x)$ has a local minimum with value of $-24$ at $x=2.$