Determine if the function$f\(\left\)(x\(\right\))$ is continuous and/or differentiable at $x=2$.

• A. Continuous and non-differentiable
• B. Continuous and differentiable
• C. Discontinuous and non-differentiable
• D. Discontinuous and differentiable

Answer: (C)

Verified step by step guidance

1. Step 1: To determine if the function is continuous at x=2, check if the left-hand limit (as x approaches 2 from the left) equals the right-hand limit (as x approaches 2 from the right) and if both equal f(2). For x < 2, f(x) = x^3, and for x ≥ 2, f(x) = (x - 2)^2.
2. Step 2: Calculate the left-hand limit of f(x) as x approaches 2 from the left. Substitute x=2 into the expression x^3 to find the value of the limit.
3. Step 3: Calculate the right-hand limit of f(x) as x approaches 2 from the right. Substitute x=2 into the expression (x - 2)^2 to find the value of the limit.
4. Step 4: Compare the left-hand limit, right-hand limit, and f(2). If all three values are equal, the function is continuous at x=2. If not, the function is discontinuous at x=2.
5. Step 5: To determine differentiability at x=2, check if the derivative of f(x) from the left-hand side equals the derivative from the right-hand side at x=2. Compute the derivative of x^3 for x < 2 and the derivative of (x - 2)^2 for x ≥ 2, then evaluate both derivatives at x=2 and compare their values.