Question

If the derivative of the function $f(x)=\{\begin{array}{ll}a{x}^2+b,& x<-1\\ b{x}^2+ax+4,& x\ge -1\end{array}$ is every where continuous, then what are the values of a and b?

• A. a = 2, b = 3
• B. a = 3, b = 2
• C. a = -2, b = -3
• D. a = -3, b = -2

Answer 1

Answer: (A)

Derivative of $\{\begin{array}{ll}a{x}^2+b,& x<-1\\ b{x}^2+ax+4,& x\ge -1\end{array}$ is
$f(x)\{\begin{array}{ll}2ax,& x<-1\\ 2bx+a,& x\ge -1\end{array}$
If $f^{\prime }(x)$ is continuous everywhere then it is also continuous at x = - 1
$f^{\prime }(x){|}_{x=-1}=-2a=-2b+a$
or $3a=2b|$ ....(i)
From the given choice $a=2,b=3$ satisfied this equation.