Find values of a and b so f in continuous at x=1 and has a derivative at x=1

Question

f(x)= {2x²+ax-1   if x≤1

        {bx³+x²+2   if x>1

 

Answer 1

For f(x) to be continuous at x = 1,

2x² +ax - 1 = bx³ + x² + 2 when x = 1.  So, a + 1 = b + 3.

 

For f(x) to be differentiable when x = 1, the function must be continuous when x = 1 and the one-sided derivatives must be equal when x = 1.

 

So, 4x + a = 3bx² + 2x  at x = 1.  Thus, 4 + a = 3b + 2.

 

We have the following system of equations:  

 

       a - b = 2

       a - 3b = -2

 

Subtract the second equation from the first to get 2b = 4.

 

So, b = 2.  Replace b by 2 in the equation a - b = 2 to get a = 4.