Find the value of k such that the function f(x) = { kx, x < 0; 0, x = 0; x^2

Find the value of k such that the function f(x) = { kx, x < 0; 0, x = 0; x^2 + k, x > 0 is continuous at x = 0.

Find the value of k such that the function f(x) = { kx, x < 0; 0, x = 0; x^2 + k, x > 0 is continuous at x = 0.

Step 1: Understand that we need to find the value of k so that the function f(x) is continuous at x = 0.
Step 2: Recall that for a function to be continuous at a point, the limit of the function as x approaches that point must equal the value of the function at that point.
Step 3: Identify the value of the function at x = 0. From the function definition, f(0) = 0.
Step 4: Determine the limit of f(x) as x approaches 0 from the left (x < 0). This is f(x) = kx. As x approaches 0, kx approaches 0. So, limit as x approaches 0 from the left is 0.
Step 5: Determine the limit of f(x) as x approaches 0 from the right (x > 0). This is f(x) = x^2 + k. As x approaches 0, x^2 approaches 0, so limit as x approaches 0 from the right is k.
Step 6: Set the two limits equal to each other for continuity: 0 (from the left) must equal k (from the right).
Step 7: Solve the equation 0 = k. This gives k = 0.
Step 8: Conclude that the value of k that makes the function continuous at x = 0 is k = 0.

Continuity of Functions – The question tests the understanding of continuity at a point, specifically how to ensure that the left-hand limit, right-hand limit, and the function value at that point are equal.
Piecewise Functions – The function is defined in pieces, and the analysis requires understanding how to evaluate limits from different sides of the point of interest.