Math Problem Statement
F(x)=x^3-3x^2+2x find tge local maxima and minima
Solution
To find the local maxima and minima of the function , we need to follow these steps:
Step 1: Find the first derivative of
The first derivative will help us find the critical points where the slope is zero.
Step 2: Set the first derivative equal to zero to find critical points
Solve this quadratic equation using the quadratic formula:
Thus, the critical points are:
Step 3: Use the second derivative to classify the critical points
The second derivative will help us determine whether each critical point is a local maximum or minimum.
Now, evaluate at each critical point:
-
For : Since , is a local minimum.
-
For : Since , is a local maximum.
Conclusion:
- Local minimum at .
- Local maximum at .
Would you like more details or further clarification?
Here are 5 related questions to explore:
- How do you find inflection points of a function?
- What is the significance of the second derivative in classifying critical points?
- Can local maxima and minima occur at endpoints of a function's domain?
- What are the global maximum and minimum values of a function?
- How does the behavior of cubic functions differ from quadratic functions in terms of critical points?
Tip: Always check the second derivative at critical points to confirm whether they are maxima or minima.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Critical Points
Second Derivative Test
Formulas
First derivative: F'(x) = 3x^2 - 6x + 2
Second derivative: F''(x) = 6x - 6
Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
Theorems
Second Derivative Test: Determines whether a critical point is a local maximum or minimum by evaluating the second derivative at that point
Suitable Grade Level
Grades 11-12
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