If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?
Practice Questions
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If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?
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To find local maxima, we first find f'(x) = 3x^2 - 6. Setting f'(x) = 0 gives x^2 - 2 = 0, so x = ±√2. Evaluating f''(x) at x = 1 gives f''(1) = 0, indicating a point of inflection. Thus, local maxima occurs at x = 1.
Questions & Step-by-step Solutions
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Q: If f(x) = x^3 - 3x^2 + 4, then the local maxima occurs at x = ?
Solution: To find local maxima, we first find f'(x) = 3x^2 - 6. Setting f'(x) = 0 gives x^2 - 2 = 0, so x = ±√2. Evaluating f''(x) at x = 1 gives f''(1) = 0, indicating a point of inflection. Thus, local maxima occurs at x = 1.
Steps: 10
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 - 6.
Step 3: Set the first derivative equal to zero to find critical points: 3x^2 - 6 = 0.
Step 4: Solve for x by simplifying the equation: x^2 - 2 = 0.
Step 5: Factor the equation to find the values of x: x = ±√2.
Step 6: Now, we need to determine if these points are local maxima or minima. We do this by finding the second derivative f''(x).
Step 7: Calculate the second derivative: f''(x) = 6x.
Step 8: Evaluate the second derivative at the critical points. First, check x = 1: f''(1) = 6(1) = 6, which is positive, indicating a local minimum.
Step 9: Check the other critical points x = √2 and x = -√2 to find the local maxima.
Step 10: Since f''(1) is positive, we conclude that the local maxima occurs at x = 1.
