Question

Find all numbers $\alpha$ for which ${\int }_1^2[{\alpha }^2+(4-4\alpha )x+4x^3]dx⩽12$

Answer 1

${\int }_1^2[{\alpha }^2+(4-4\alpha )x+4x^3]dx\le 12$

$\Rightarrow {\int }_1^2{\alpha }^{2}dx+{\int }_1^{2}4xdx+{\int }_1^2(-4\alpha )xdx+4{\int }_1^2{x}^{3}dx\le 12$

$\Rightarrow {\alpha }^2+\frac{4x^2}{2}{|}_1^2+(-4\alpha )\frac{x^2}{2}{|}_1^2+\frac{4x^4}{4}{|}_1^2\le 12$

$\Rightarrow {\alpha }^2+2\left[3\right]-2\alpha \left(3\right)+(2^4-1)\le 12$

$\Rightarrow {\alpha }^2-6\alpha +2^4+5\le 12$

$\Rightarrow {\alpha }^2-6\alpha +9\le 0$

$\Rightarrow (\alpha -3{)}^2\le 0$

$(\alpha -3{)}^2$ will always be $0$ greater than zero except at $\alpha =3$,

at $\alpha =3,(\alpha -3)=0$.