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For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflecti
For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
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Practice Questions
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Q1
For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
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f''(x) = 18x - 24. Setting f''(x) = 0 gives x = 4/3. This is the inflection point.
Questions & Step-by-step Solutions
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Q
Q: For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
Solution:
f''(x) = 18x - 24. Setting f''(x) = 0 gives x = 4/3. This is the inflection point.
Steps: 7
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Step 1: Start with the function f(x) = 3x^3 - 12x^2 + 9.
Step 2: Find the first derivative f'(x) to understand the slope of the function.
Step 3: Calculate the second derivative f''(x) to find 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 18x - 24 = 0 for x.
Step 6: Rearrange the equation to find x = 24/18, which simplifies to x = 4/3.
Step 7: Conclude that the x-coordinate of the inflection point is x = 4/3.
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