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

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

Options:

Correct Answer: (1, 3)

Solution:

f\'\'(x) = 12x^2 - 24x. Setting f\'\'(x) = 0 gives x = 0 and x = 2. The point of inflection is at (1, 3).

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

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

Step 1: Start with the function f(x) = x^4 - 4x^3 + 6.
Step 2: Find the first derivative f'(x) to understand the slope of the function.
Step 3: Calculate f'(x) = 4x^3 - 12x^2.
Step 4: Find the second derivative f''(x) to determine the concavity of the function.
Step 5: Calculate f''(x) = 12x^2 - 24x.
Step 6: Set the second derivative equal to zero: 12x^2 - 24x = 0.
Step 7: Factor the equation: 12x(x - 2) = 0.
Step 8: Solve for x: This gives x = 0 and x = 2.
Step 9: To find the point of inflection, we need to check the function value at x = 1 (the midpoint between 0 and 2).
Step 10: Calculate f(1) = 1^4 - 4(1^3) + 6 = 1 - 4 + 6 = 3.
Step 11: The point of inflection is at (1, 3).

Second Derivative Test – The point of inflection occurs where the second derivative changes sign, which is found by setting the second derivative equal to zero.
Finding Points of Inflection – To find points of inflection, you need to determine where the second derivative is zero and check for sign changes.