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For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection
For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.
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Practice Questions
1 question
Q1
For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.
(0, 16)
(2, 0)
(4, 0)
(2, 4)
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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).
Questions & Step-by-step Solutions
1 item
Q
Q: For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.
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).
Steps: 9
Show Steps
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).
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