Free response Q in AP calculus AB Plase help me?

Question

1. Let f be the function defined as
f(x)= absoulute val of(x-1) +2 for x<1
ax^2+bx for x greater or equal to 1
where a and b are constants.
(a) If a = 2 and b = 3, is f continuous for all x? Justify your answer.

(b) Describe all values of a and b for which f is a continuous function.

(c) For what values of a and b is f both continuous and differentiable.

plz add ore explanation for (b)

Answer 1

1. Let f be the function defined as
f(x) = |x - 1| + 2, for x < 1
f(x) = ax² + bx, for x ≥ 1
where a and b are constants.

(a) If a = 2 and b = 3, is f continuous for all x? Justify your answer.

If f is continuous, then at x = 1 both definitions of f(1) should be the same. There is no potential problem elsewhere.

f(1) = |1 - 1| + 2 = 0 + 2 = 2
f(1) = a*1² + b*1 = a + b = 2 + 3 = 5

They are different so the function is not continuous at x = 1.

(b) Describe all values of a and b for which f is a continuous function.

a + b = 2

(c) For what values of a and b is f both continuous and differentiable.

For this to be true, f(x) needs to be the same for both functions at x = 1 and so does f'(x).

f(x) = |x - 1| + 2, for x < 1
f'(x) = -1

f(x) = ax² + bx, for x ≥ 1
f'(x) = 2ax + b = -1
f'(1) = 2a + b = -1

Putting the two equations together we get:

a + b = 2
2a + b = -1

Subtract the second equation from the first.

-a = 3
a = -3

Substitute back into the first equation to solve for b.

a + b = 2
-3 + b = 2
b = 5

For a = -3 and b = 5, f is both continuous and differentiable.