Question

Consider the function, $F\left(x\right)={\int }_{-1}^x(t^2-t)dt,x\in R$. Find the $x$ and $y$ intercept of $F\left(x\right)$ if it exist. $F(-1)=0$

• A. $(-1,0),(0,5/6)$
• B. $(1,0),(0,-5/6)$
• C. $(5/6,0),(0,1)$
• D. $(5/6,0),(0,-1)$

Answer 1

The correct option is A $(-1,0),(0,5/6)$

$F\left(x\right)={\int }_{-1}^x(t^2-t)dt$
Using Leibniz rule of differentiation,
$F^{\prime }\left(x\right)=(x^2-x)\frac{d}{dx}x=(x^2-x)$
Integrating we get,
$\Rightarrow F\left(x\right)=\frac{x^3}{3}-\frac{x^2}{2}+c$
Also $F(-1)=0\Rightarrow c=\frac{5}{6}$
Thus $F\left(x\right)=\frac{1}{6}(2x^3-3x^2+5)$
Hence intercepts on the axes are $(-1,0),(0,5/6)$