For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f

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Question: For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.

Options:

Correct Answer: 1

Solution:

Setting f(1-) = f(1+) and f\'(1-) = f\'(1+) gives k = 2 for differentiability.

For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.

For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.

Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 for x < 1 and f(x) = kx + 1 for x >= 1.
Step 2: To ensure f is differentiable at x = 1, the function must be continuous at that point. This means f(1-) must equal f(1+).
Step 3: Calculate f(1-) using the first part of the function: f(1-) = (1)^2 = 1.
Step 4: Calculate f(1+) using the second part of the function: f(1+) = k(1) + 1 = k + 1.
Step 5: Set f(1-) equal to f(1+): 1 = k + 1.
Step 6: Solve for k: k = 1 - 1 = 0.
Step 7: Next, we need to check the derivatives to ensure differentiability. Calculate f'(x) for both parts of the function.
Step 8: For x < 1, f'(x) = 2x. Therefore, f'(1-) = 2(1) = 2.
Step 9: For x >= 1, f'(x) = k. Therefore, f'(1+) = k.
Step 10: Set f'(1-) equal to f'(1+): 2 = k.
Step 11: Solve for k: k = 2.
Step 12: Therefore, the value of k that makes f differentiable at x = 1 is k = 2.

Piecewise Functions – Understanding how to evaluate and differentiate functions defined in pieces.
Continuity and Differentiability – Knowing that for a function to be differentiable at a point, it must also be continuous at that point.
Limits and Derivatives – Applying the concept of limits to find the left-hand and right-hand derivatives at a specific point.