For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection

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Question: For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.

Options:

Correct Answer: (2, 0)

Solution:

Find f\'\'(x) = 12x^2 - 16. Setting f\'\'(x) = 0 gives x^2 = 4, so x = ±2. f(2) = 0, thus the inflection point is (2, 0).

For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.

For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.

Correct Answer: (2, 0)

Step 1: Start with the function f(x) = x^4 - 8x^2 + 16.
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 inflection points.
Step 5: Solve the equation 12x^2 - 16 = 0 to find the values of x.
Step 6: Simplify the equation to get x^2 = 4, which gives x = ±2.
Step 7: Calculate the function value f(2) to find the y-coordinate of the inflection point.
Step 8: Substitute x = 2 into the original function: f(2) = 2^4 - 8(2^2) + 16 = 0.
Step 9: The coordinates of the inflection point are (2, 0).

Second Derivative Test – The second derivative is used to determine the concavity of a function and to find inflection points where the concavity changes.
Finding Inflection Points – Inflection points occur where the second derivative is zero or undefined, indicating a change in concavity.
Evaluating Functions – To find the coordinates of the inflection point, the original function must be evaluated at the x-values found from the second derivative.