If f(x) = { x^2, x < 0; kx + 1, x >= 0 } is differentiable at x = 0, what

If f(x) = { x^2, x < 0; kx + 1, x >= 0 } is differentiable at x = 0, what is k?

If f(x) = { x^2, x < 0; kx + 1, x >= 0 } is differentiable at x = 0, what is k?

Step 1: Understand that the function f(x) is defined in two parts: f(x) = x^2 when x is less than 0, and f(x) = kx + 1 when x is greater than or equal to 0.
Step 2: To find the value of k that makes f(x) differentiable at x = 0, we need to ensure that both the function and its derivative are continuous at that point.
Step 3: First, check the value of f(0) using the second part of the function: f(0) = k(0) + 1 = 1.
Step 4: Now, find the derivative of f(x) for both parts. The derivative of f(x) = x^2 is f'(x) = 2x for x < 0. The derivative of f(x) = kx + 1 is f'(x) = k for x >= 0.
Step 5: Evaluate the derivative at x = 0 from the left side: f'(0-) = 2(0) = 0.
Step 6: Evaluate the derivative at x = 0 from the right side: f'(0+) = k.
Step 7: For f(x) to be differentiable at x = 0, the left-hand derivative must equal the right-hand derivative: 0 = k.
Step 8: Therefore, we conclude that k must be 0.

Piecewise Functions – Understanding how to analyze functions defined by different expressions based on the input value.
Differentiability – Knowing that a function is differentiable at a point if it is continuous there and the derivatives from both sides are equal.
Finding Derivatives – Calculating the derivative of each piece of the function and setting them equal to ensure differentiability at the specified point.