For which value of k is the function f(x) = kx^2 + 2x + 1 differentiable at x =

For which value of k is the function f(x) = kx^2 + 2x + 1 differentiable at x = -1?

For which value of k is the function f(x) = kx^2 + 2x + 1 differentiable at x = -1?

Step 1: Understand that a function is differentiable at a point if its derivative exists at that point.
Step 2: Identify the function given: f(x) = kx^2 + 2x + 1.
Step 3: Find the derivative of the function f(x). The derivative f'(x) is calculated as follows: f'(x) = 2kx + 2.
Step 4: Substitute x = -1 into the derivative to find f'(-1): f'(-1) = 2k(-1) + 2 = -2k + 2.
Step 5: Recognize that for the function to be differentiable at x = -1, f'(-1) must exist for any value of k.
Step 6: Since -2k + 2 is a linear expression, it exists for all values of k. Therefore, the function is differentiable at x = -1 for any k.

Differentiability – The property of a function to have a derivative at a given point, which requires the function to be continuous and smooth at that point.
Finding Derivatives – The process of calculating the derivative of a function, which involves applying rules of differentiation to obtain the slope of the function at any point.