If f(x) = x^4 - 8x^2 + 16, then the points of inflection are at:

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Question: If f(x) = x^4 - 8x^2 + 16, then the points of inflection are at:

Options:

Correct Answer: x = ±2

Solution:

To find points of inflection, we find f\'\'(x) = 12x^2 - 16. Setting f\'\'(x) = 0 gives x^2 = 4, so x = ±2 are points of inflection.

If f(x) = x^4 - 8x^2 + 16, then the points of inflection are at:

If f(x) = x^4 - 8x^2 + 16, then the points of inflection are at:

Step 1: Start with the function f(x) = x^4 - 8x^2 + 16.
Step 2: Find the first derivative f'(x) to determine the slope of the function.
Step 3: Calculate f'(x) = 4x^3 - 16x.
Step 4: Find the second derivative f''(x) to determine the concavity of the function.
Step 5: Calculate f''(x) = 12x^2 - 16.
Step 6: Set the second derivative equal to zero: 12x^2 - 16 = 0.
Step 7: Solve for x by adding 16 to both sides: 12x^2 = 16.
Step 8: Divide both sides by 12: x^2 = 16/12, which simplifies to x^2 = 4/3.
Step 9: Take the square root of both sides to find x: x = ±2.
Step 10: The points of inflection are at x = 2 and x = -2.

Second Derivative Test – Understanding how to find points of inflection by analyzing the second derivative of a function.
Critical Points – Identifying where the second derivative equals zero to find potential points of inflection.
Behavior of Functions – Recognizing how changes in concavity relate to the second derivative.