Question: For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.
Options:
(1, 1)
(2, 2)
(3, 3)
(4, 4)
Correct Answer: (2, 2)
Solution:
f\'\'(x) = 12x - 18. Setting f\'\'(x) = 0 gives x = 1.5. The inflection point is (1.5, f(1.5)).
For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.
Practice Questions
Q1
For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.
(1, 1)
(2, 2)
(3, 3)
(4, 4)
Questions & Step-by-Step Solutions
For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.
Correct Answer: (1.5, f(1.5))
Step 1: Start with the function f(x) = 2x^3 - 9x^2 + 12x.
Step 2: Find the first derivative f'(x) to determine 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 from Step 4 to find the x-value of the inflection point.
Step 6: Substitute the x-value found in Step 5 back into the original function f(x) to find the corresponding y-value.
Step 7: Combine the x and y values to write the inflection point as (x, y).
Second Derivative Test – The inflection point is found by setting the second derivative equal to zero and solving for x.
Function Evaluation – After finding the x-coordinate of the inflection point, the corresponding y-coordinate must be calculated by substituting x back into the original function.
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