If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?
Practice Questions
1 question
Q1
If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?
x = 0
x = 1
x = 2
x = 3
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.
Questions & Step-by-step Solutions
1 item
Q
Q: If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at which point?
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.
Steps: 9
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.
