Find the values of a and b that make 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

The two pieces of the graph must fit together when x = 1 and the one-sided derivatives must agree

when x = 1.

So, 1² - 2(1) - a = -(1)² + b + 2 and 2x - 2 = -2x + b when x = 1

We get: -1 - a = 1 + b and 0 = -2 + b

b = 2 and a = -4