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

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

Options:

Correct Answer: 1

Solution:

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

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

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

Step 1: Understand that a function is differentiable at a point if its derivative exists at that point.
Step 2: Find the derivative of the function f(x) = x^2 - ax + 2.
Step 3: The derivative f'(x) is calculated as follows: f'(x) = 2x - a.
Step 4: Substitute x = 1 into the derivative to find f'(1): f'(1) = 2(1) - a = 2 - a.
Step 5: For the function to be differentiable at x = 1, we need to set the derivative equal to 0: 2 - a = 0.
Step 6: Solve the equation 2 - a = 0 for a: a = 2.
Step 7: Therefore, the value of a that makes the function differentiable at x = 1 is a = 2.

Differentiability – The function must have a defined derivative at the point of interest, which in this case is x = 1.
Finding Derivatives – Calculating the derivative of the function and evaluating it at a specific point.
Setting Derivative to Zero – Understanding that for a function to be differentiable at a point, the derivative can be set to zero to find critical points.