For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -

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Question: For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -1?

Options:

Correct Answer: 0

Solution:

To ensure differentiability at x = -1, we find f\'(-1) exists. Setting a = 0 ensures the derivative is defined.

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

For which value of a is the function f(x) = x^2 + ax + 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) = x^2 + ax + 1.
Step 3: Find the derivative of the function, f'(x). The derivative of f(x) is f'(x) = 2x + a.
Step 4: Substitute x = -1 into the derivative to find f'(-1). This gives f'(-1) = 2(-1) + a = -2 + a.
Step 5: For the function to be differentiable at x = -1, f'(-1) must exist. This means we need to find a value for a.
Step 6: Set the expression -2 + a equal to a specific value. If we set a = 2, then f'(-1) = -2 + 2 = 0, which is defined.
Step 7: Conclude that the function f(x) is differentiable at x = -1 when a = 2.

Differentiability – A function is differentiable at a point if its derivative exists at that point.
Finding Derivatives – To check differentiability, we need to compute the derivative of the function and evaluate it at the given point.
Continuity – A function must be continuous at a point for it to be differentiable there.