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In a function f(x) = x^3 - 3x, what is the nature of the critical points?
In a function f(x) = x^3 - 3x, what is the nature of the critical points?
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Practice Questions
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Q1
In a function f(x) = x^3 - 3x, what is the nature of the critical points?
All critical points are local maxima.
All critical points are local minima.
There are both local maxima and minima.
There are no critical points.
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The function has critical points where the first derivative is zero, which can be analyzed to find both local maxima and minima.
Questions & Step-by-step Solutions
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Q
Q: In a function f(x) = x^3 - 3x, what is the nature of the critical points?
Solution:
The function has critical points where the first derivative is zero, which can be analyzed to find both local maxima and minima.
Steps: 5
Show Steps
Step 1: Find the first derivative of the function f(x) = x^3 - 3x.
Step 2: Set the first derivative equal to zero to find critical points.
Step 3: Solve the equation from Step 2 to find the values of x where the critical points occur.
Step 4: Use the second derivative test to determine the nature of each critical point (local maximum, local minimum, or saddle point).
Step 5: Analyze the results from Step 4 to conclude the nature of the critical points.
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