In a function f(x) = x^3 - 3x, what is the nature of the critical points?

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Question: In a function f(x) = x^3 - 3x, what is the nature of the critical points?

Options:

Correct Answer: There are both local maxima and minima.

Solution:

The function has critical points where the first derivative is zero, which can be analyzed to find both local maxima and minima.

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?

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.

Critical Points – Points where the first derivative of the function is zero or undefined, indicating potential local maxima, minima, or points of inflection.
First Derivative Test – A method used to determine the nature of critical points by analyzing the sign of the first derivative before and after the critical points.
Second Derivative Test – An alternative method to classify critical points by evaluating the second derivative at those points to determine concavity.