Determine the point of inflection for the function f(x) = x^4 - 4x^3 + 6x^2.

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Question: Determine the point of inflection for the function f(x) = x^4 - 4x^3 + 6x^2.

Options:

Correct Answer: (1, 3)

Solution:

Find f\'\'(x) = 12x^2 - 24x + 12. Setting f\'\'(x) = 0 gives x = 1 and x = 2. Testing intervals shows a change in concavity at x = 1.

Determine the point of inflection for the function f(x) = x^4 - 4x^3 + 6x^2.

Determine the point of inflection for the function f(x) = x^4 - 4x^3 + 6x^2.

Step 1: Start with the function f(x) = x^4 - 4x^3 + 6x^2.
Step 2: Find the first derivative f'(x) to understand the slope of the function.
Step 3: Find the second derivative f''(x) to determine the concavity of the function.
Step 4: Set the second derivative f''(x) equal to 0 to find potential points of inflection.
Step 5: Solve the equation 12x^2 - 24x + 12 = 0 to find the values of x.
Step 6: Factor or use the quadratic formula to find the solutions, which are x = 1 and x = 2.
Step 7: Test intervals around x = 1 and x = 2 to see if the concavity changes.
Step 8: Determine that there is a change in concavity at x = 1, confirming it as a point of inflection.

Second Derivative Test – The question tests the understanding of finding points of inflection using the second derivative of a function.
Concavity – It assesses the ability to determine changes in concavity by analyzing the sign of the second derivative.
Critical Points – The question involves identifying critical points where the second derivative equals zero.