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Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.
Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.
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Practice Questions
1 question
Q1
Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.
(1, 3)
(2, 2)
(3, 1)
(0, 6)
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f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. The point of inflection is at (2, f(2)) = (2, 2).
Questions & Step-by-step Solutions
1 item
Q
Q: Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.
Solution:
f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. The point of inflection is at (2, f(2)) = (2, 2).
Steps: 11
Show Steps
Step 1: Start with the function f(x) = x^4 - 4x^3 + 6.
Step 2: Find the first derivative f'(x) to determine 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 or x = 2.
Step 9: To find the point of inflection, we need to find the corresponding y-value for x = 2.
Step 10: Calculate f(2) = (2)^4 - 4(2)^3 + 6 = 16 - 32 + 6 = -10.
Step 11: The point of inflection is at (2, -10).
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