Find the value of k for which the function f(x) = x^3 - 3kx^2 + 3k^2x - k^3 is d

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Question: Find the value of k for which the function f(x) = x^3 - 3kx^2 + 3k^2x - k^3 is differentiable at x = k.

Options:

Correct Answer: 1

Solution:

For f(x) to be differentiable at x = k, f\'(k) must exist. Setting k = 1 makes f\'(k) continuous.

Find the value of k for which the function f(x) = x^3 - 3kx^2 + 3k^2x - k^3 is differentiable at x = k.

Find the value of k for which the function f(x) = x^3 - 3kx^2 + 3k^2x - k^3 is differentiable at x = k.

Step 1: Understand that for a function to be differentiable at a point, it must be smooth and continuous at that point.
Step 2: Identify the function given: f(x) = x^3 - 3kx^2 + 3k^2x - k^3.
Step 3: Find the derivative of the function, f'(x), using the power rule.
Step 4: Calculate f'(x) = 3x^2 - 6kx + 3k^2.
Step 5: Substitute x = k into the derivative to find f'(k). This gives f'(k) = 3k^2 - 6k^2 + 3k^2 = 0.
Step 6: Check if f(k) is continuous by substituting k into the original function f(x).
Step 7: Set k = 1 to see if it makes f'(k) continuous and differentiable.
Step 8: Verify that with k = 1, the function behaves well at x = k.

Differentiability – The function must have a defined derivative at the point of interest, which in this case is x = k.
Polynomial Functions – Understanding the properties of polynomial functions, which are differentiable everywhere.
Critical Points – Identifying points where the derivative is zero or undefined to analyze behavior around x = k.