Let the function, f(x)=-3ax^2-2, & xbe differentiable for all x R, where a1,b R.

Question

Let the function,

$f(x)=-3{ax}^{2}-2, \& xbe differentiable for all $x\in R$, where $a>1,b\in R$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha +\beta \sqrt{3},\alpha ,\beta \in Z$, then the value of $

Accepted Answer

$f(x)$ is continuous and differentiable

$\begin{gathered} x=1; == \ -3a-2={a}^{2}+b,-6a=b \ a=2,1;b=-12 \end{gathered}$

$f(x)= $$-6{x}^{2}-2, \& x$$

$$$\begin{gathered} ={\int }_{1}^{-\sqrt{3}}(-6{x}^{2}-2+20)dx+{\int }_{2}^{1}(4-12x+20)dx] \ =16+12\sqrt{3}+6=22+12\sqrt{3} \ \therefore \, \alpha +\beta =34 \end{gathered}$$

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