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For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k
For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?
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For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?
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Setting the derivative f'(-1) = 0 gives k = 1 for differentiability.
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Q: For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?
Solution:
Setting the derivative f'(-1) = 0 gives k = 1 for differentiability.
Steps: 7
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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.
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