The values a = 1/4 and b = 1 make f'(4) exist.
How the values of a and b were calculated.
For f'(4) to exist, the two pieces of the function f(x) must be differentiable at x = 4
For f(x) = √x when x <=4
f'(x) = 1/2√x
2. For f(x) = ax + b when x > 4
f'(x) = a
For f'(4) to exist, set the derivatives equal to each other at x = 4 and solve for a
1/2√4 = a
1/4 * 4 = a²
a² = √1/16 = 1/4
The value of a is 1/4
To find b, f(x) must be continuous at x = 4.
So, equate the two pieces of the function at x = 4
√4 = a(4) + b
Solve this equation for b
2 = 1/4 * 4 + b
2 = 1 + b
b = 1
So, the values a = 1/4 and b = 1 make f'(4) exist.