If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

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Question: If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

Options:

x = 0
x = 1
x = 2
x = 3

Correct Answer: x = 1

Solution:

To find local maxima, we first find f\'(x) = 3x^2 - 6x. Setting f\'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2. Checking the second derivative f\'\'(x) = 6x - 6, we find f\'\'(2) < 0, indicating a local maxima at x = 2.

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

Practice Questions

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

x = 0
x = 1
x = 2
x = 3

Questions & Step-by-Step Solutions

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

Step 1: Start with the function f(x) = x^3 - 3x^2 + 4.
Step 2: Find the first derivative f'(x) to determine where the slope is zero. The first derivative is f'(x) = 3x^2 - 6x.
Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 6x = 0.
Step 4: Factor the equation: 3x(x - 2) = 0.
Step 5: Solve for x: This gives us two solutions, x = 0 and x = 2.
Step 6: To determine if these points are local maxima or minima, we need to find the second derivative f''(x). The second derivative is f''(x) = 6x - 6.
Step 7: Evaluate the second derivative at the critical points. First, check x = 0: f''(0) = 6(0) - 6 = -6 (which is less than 0, indicating a local maxima).
Step 8: Now check x = 2: f''(2) = 6(2) - 6 = 6 (which is greater than 0, indicating a local minima).
Step 9: Conclude that the local maxima occurs at x = 0.

Finding Local Maxima – The process of determining points where a function reaches a local maximum by using first and second derivatives.
Critical Points – Identifying points where the first derivative is zero or undefined to find potential local maxima or minima.
Second Derivative Test – Using the second derivative to determine the concavity of the function at critical points to classify them as local maxima or minima.

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

What’s inside this PDF?

If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?

Practice Questions

Questions & Step-by-Step Solutions

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