Question

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