Question

$If{\int }_0^1{e}^{x^2}(x-\alpha )dx=0,then$

• A. $1<\alpha <2$
• B. $\alpha <0$
• C. $0<\alpha <1$
• D. $\alpha =0$

Answer 1

The correct option is C $0<\alpha <1$

$\because {\int }_0^1{e}^{x^2}(x-\alpha )dx=0,$
$\therefore f\left(x\right)=e^{x^2}(x-\alpha )$ must cross x-axis between
x = 0 and x = 1.
Now f(x) = 0 at x =$\alpha$
Since 0 < x < 1,
$\therefore 0<\alpha <1$