Let f: ℝ → ℝ be a function. Note: ℝ denotes the set of real numbers. f(x) =…
2024
Let f: ℝ → ℝ be a function. Note: ℝ denotes the set of real numbers.
f(x) = -x, if x < -2;
f(x) = ax² + bx + c, if x ∈ [-2, 2];
f(x) = x, if x > 2.
Which ONE of the following choices gives the values of a, b, c that make the function f continuous and differentiable?
- A.
a = 1/4, b = 0, c = 1
- B.
a = 1/2, b = 0, c = 0
- C.
a = 0, b = 0, c = 0
- D.
a = 1, b = 1, c = -4
Show answer & explanation
Correct answer: A
Key conditions: ensure continuity and differentiability at x = -2 and x = 2.
Continuity at x = -2: left piece gives f(-2-) = -(-2) = 2, so 4a - 2b + c = 2.
Differentiability at x = -2: derivative from left is -1, derivative of ax^2+bx+c is 2ax + b, so -4a + b = -1.
Continuity at x = 2: right piece gives f(2+) = 2, so 4a + 2b + c = 2.
Differentiability at x = 2: derivative from right is 1, so 4a + b = 1.
Solving the system: subtract the two continuity equations to get 4b = 0, so b = 0. Then 4a + b = 1 gives 4a = 1, so a = 1/4. Finally 4a + 2b + c = 2 gives 1 + 0 + c = 2, so c = 1.
Conclusion: a = 1/4, b = 0, c = 1 satisfy all continuity and differentiability conditions.