Question

The point of inflection of the curve $y=x^4$ is at

• A. $x=0$
• B. $x=3$
• C. $x=12$
• D. No where

Answer 1

The correct option is D No where

Given $y=x^4$

$\Rightarrow y^{\prime }=4x^3$

$\Rightarrow y^{\prime \prime }=12x^2=0$ for $x=0$

Hence, $x=0$ is a possible point of inflection, but we need to verify. So, check in the neighborhood of $0$,

$f^{\prime \prime }\left(0^{+}\right)=12(0^{+}{)}^2>0$ and $f^{\prime \prime }\left(0^{-}\right)=12(0^{-}{)}^2>0$

So above curve is concave upward for all real numbers

Hence there is no point of inflection for given curve