For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k

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Question: For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?

Options:

Correct Answer: 1

Solution:

Setting the derivative f\'(-1) = 0 gives k = 1 for differentiability.

For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?

For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?

Step 1: Understand that for a function to be differentiable at a point, it must be continuous and have a defined derivative at that point.
Step 2: Write down the function: f(x) = x^2 + kx + 1.
Step 3: Find the derivative of the function, f'(x). The derivative of f(x) is f'(x) = 2x + k.
Step 4: Substitute x = -1 into the derivative to find f'(-1). This gives f'(-1) = 2(-1) + k = -2 + k.
Step 5: For the function to be differentiable at x = -1, we need f'(-1) to equal 0. So, set -2 + k = 0.
Step 6: Solve the equation -2 + k = 0 for k. This gives k = 2.
Step 7: Conclude that for the function f(x) to be differentiable at x = -1, k must be 2.

Differentiability – The condition for a function to be differentiable at a point requires that the derivative exists at that point.
Finding Derivatives – Calculating the derivative of a polynomial function to analyze its behavior at specific points.
Setting Derivatives to Zero – Understanding that for a function to have a local extremum or to check for differentiability, the derivative can be set to zero.