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

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

differentiating with respect to x

$f'(x)=12x^3-12x^2-24x$

to find stationary points put $f'(x)=0$

$\Rightarrow12x^3-12x^2-24x=0$

$\Rightarrow x^3-x^2-2x=0$

$\Rightarrow x(x^2-x-2)=0$

$\Rightarrow x(x-2)(x+1)=0$

stationary points are

$x=-1,0,2$

differentiating again with respect to x

$f''(x)=36x^2-24x-24$

$f''(-1)=36>0 \space(x=-1 \space is \space local\space minimum)$

$f''(0)=-24<0\space(x=0 \space is \space local \space maximum)$

$f''(2)=72>0\space(x=2 \space is \space local\space minimum)$