Question

Find the inflection points and the intervals in which the function $f\left(x\right)=x^4-4x^3$ is concave up and concave down.

Answer 1

$f\left(x\right)=x^4-4x^3$

$\Rightarrow f^{\prime }\left(x\right)=4x^3-12x^2$

$\Rightarrow f^{\prime \prime }\left(x\right)=12x^2-24x$$=12x\left(x-2\right)$

for point of inflection

$f^{\prime \prime }\left(x\right)=0$

$\Rightarrow 12x\left(x-2\right)=0$

$\Rightarrow x=0,2$

So, $x=0,2$ are the points of inflection

For concave upward

$f^{\prime \prime }\left(x\right)>0$

$\Rightarrow 12x\left(x-2\right)>0$

$\Rightarrow x\in \left(-\infty ,0\right)\cup \left(2,\infty \right)$

For concave downward

$f^{\prime \prime }\left(x\right)<0$

$\Rightarrow 12x\left(x-2\right)<0$

$\Rightarrow x\in \left(0,2\right)$