Find the value of c such that the function f(x) = { x^2 + c, x < 1; 2x + 1, x

Find the value of c such that the function f(x) = { x^2 + c, x < 1; 2x + 1, x >= 1 } is differentiable at x = 1.

Find the value of c such that the function f(x) = { x^2 + c, x < 1; 2x + 1, x >= 1 } is differentiable at x = 1.

Step 1: Understand that the function f(x) has two parts: one for x < 1 and another for x >= 1.
Step 2: For the function to be differentiable at x = 1, it must be continuous at that point.
Step 3: Find the value of f(1) using the second part of the function (since x = 1 falls in that category): f(1) = 2(1) + 1 = 3.
Step 4: Now, find the limit of f(x) as x approaches 1 from the left (x < 1): f(1) = 1^2 + c = 1 + c.
Step 5: Set the left-hand limit equal to the right-hand limit: 1 + c = 3.
Step 6: Solve for c: c = 3 - 1 = 2.
Step 7: Check the derivatives from both sides to ensure differentiability: The derivative from the left is 2(1) = 2, and from the right is 2. Since both are equal, the function is differentiable.

Piecewise Functions – Understanding how to analyze functions defined by different expressions over different intervals.
Continuity and Differentiability – Knowing that for a function to be differentiable at a point, it must also be continuous at that point.
Limits – Using limits to find the values of functions as they approach a certain point from both sides.