Question

Let $y=a{x}^n,a\ne 0$ has a point of inflection at $x=0,$ then which of the following is correct

• A. $n\in \text{odd integers}$
• B. $n\in \text{odd integers},n\ne 1$
• C. $n\in \text{even integers}$
• D. $n\in \text{even integers},n\ne 2$

Answer 1

The correct option is B $n\in \text{odd integers},n\ne 1$

Given curve : $y=a{x}^n$
$\Rightarrow y^{\prime \prime }=n(n-1)a{x}^{n-2}$
for point of inflection at $x=0:$
$y^{\prime \prime }\left(0^{+}\right)\cdot y^{\prime \prime }\left(0^{-}\right)<0$
only possible if, $n-2\in \text{odd integers}$ and $n-2\ne 0$
$\Rightarrow n\in \text{odd integers}$
but when $n=1,y^{\prime \prime }\left(0^{+}\right)=y^{\prime \prime }\left(0^{-}\right)$
So, $n\in \text{odd integers}$ except one.