Question

The number of distinct inflection points of $f\left(x\right)=x^4-4x^3+12x^2+x-1$ is

Answer 1

$f\left(x\right)=x^4-4x^3+12x^2+x-1$
$\Rightarrow f^{\prime }\left(x\right)=4x^3-12x^2+24x+1$
$\Rightarrow f^{\prime \prime }\left(x\right)=12x^2-24x+24$
$=12(x^2-2x+2)$
$=\left[\right(x-1{)}^2+1]>0$ $\forall x\in R$
Hence, $f\left(x\right)$ is always concave upwards. So, number of inflection point $=0$