Question

Given that the derivative of $f\left(x\right)=\{\begin{array}{cc}A{x}^3+Bx+2& x\le 2\\ B{x}^2-A& x>2\end{array}$ is continuous for all x. The values of A and B (respectively) are

• A. -2,-8
• B. -1,-6
• C. -2,-4
• D. 2,4

Answer 1

The correct option is A -2,-8

$f\left(x\right)=\{\begin{array}{cc}A{x}^3+Bx+2& x\le 2\\ B{x}^2-A& x>2\end{array}$
given derivative of $f\left(x\right)$ is continuous everywhere $\Rightarrow f\left(x\right)$ is continuous everywhere
$f^{\prime }\left(x\right)=\{\begin{array}{ccc}3A{x}^2+B& x\le 22Bx& x>2\end{array}$
$f^{\prime }\left(2^{+}\right)=f^{\prime }\left(2^{-}\right)\Rightarrow 4A=B$
also$f\left(2\right)=f\left(2\right)$$\Rightarrow 9A-2B=-2$
$\Rightarrow A=-2,B=-8$