For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflecti
Practice Questions
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For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
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Questions & Step-by-Step Solutions
For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
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.
Second Derivative Test – The second derivative of a function is used to determine the concavity and identify 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.
