Question

Consider the function, $F\left(x\right)={\int }_{-1}^x(t^2-t)dt,x\in R$. Find the inflection point.

• A. $x=\frac{1}{2}$
• B. $x=1$
• C. $x=0$
• D. inflection point not exists

Answer 1

The correct option is A $x=\frac{1}{2}$

Given, $F\left(x\right)={\int }_{-1}^x(t^2-t)dt$
Using Leibniz rule of differentiation,
$F^{\prime }\left(x\right)=(x^2-x)\frac{d}{dx}x=(x^2-x)$
$\Rightarrow F^{\prime \prime }\left(x\right)=2x-1$
Now for inflection point $f^{\prime \prime }\left(x\right)=0\Rightarrow x=\frac{1}{2}$
Hence $x=\frac{1}{2}$ is inflection point of F.