calculus, differentiable function

Question

Find b , if it is known that the function f(x)={x+3 if x<3 ; ax²+bx if x>=3 is differentiable, where a and b are parameters.

Answer 1

The given function is f(x) = {x + 3 if x < 3; ax² + bx if x ≥ 3.

For this function to be continuous, we must have that

limit x→3^- f(x) = limit x→3⁺ f(x)

(3) + 3 = a(3)² + b(3)

6 = 9a + 3b

or

4 = 6a + 2b ---(1)

The derivative of this function is f'(x) = {1 if x < 3; 2ax + b if x ≥ 3.

To be differentiable, we must have that

limit x→3^- f'(x) = limit x→3⁺ f'(x)

1 = 2a(3) + b

1 = 6a + b ---(2)

Subtract equation (2) from equation (1) to get

4 - 1 = 6a - 6a + 2b - b

b = 3