Let the function $f(x)=\begin{cases}-3ax^2-2,&x1$, $b\in\mathbf R$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta\sqrt3$, $\alpha,\beta\in\mathbf Z$, then the value of $\alpha+\beta$ is _______

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Continuity at 1: $-3a-2=a^2+b$; differentiability: $-6a=b$. So $a^2-3a+2=0$, $a=2$ and $b=-12$. $f(x)=-20$ at $x=-\sqrt3$ and $x=2$. Area $=\int_{-\sqrt3}^{1}(18-6x^2)\,dx+\int_1^2(24-12x)\,dx=(16+12\sqrt3)+6=22+12\sqrt3$. So $\alpha+\beta=34$. Answer: 34.

Explanation

Reviewed by Dr. Abubacker

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