Find the values of a and b that make the function f(x) continuous and differentiable at x=1.

Question

f(x) = {x^2 - 2x - a, x<1

          

          {-x^2 + bx +2, x>=1

 

Answer 1

To be continuous at x = 1, we must have x² - 2x - a = -x² + bx + 2 when x = 1.

 

               So, -1 - a = 1 + b.  Therefore, a + b = -2

 

To be differentiable at x = 1, we need the "one-sided" derivatives to be the same at x = 1.

 

               So, 2x - 2 = -2x + b when x = 1.

 

                            b = 2 and a = -4