If a function f(x) is defined as f(x) = x^3 - 3x, what is the nature of its crit
Practice Questions
Q1
If a function f(x) is defined as f(x) = x^3 - 3x, what is the nature of its critical points?
They can be local maxima, local minima, or points of inflection.
They are always local maxima.
They are always local minima.
They do not exist.
Questions & Step-by-Step Solutions
If a function f(x) is defined as f(x) = x^3 - 3x, what is the nature of its critical points?
Step 1: Find the derivative of the function f(x) = x^3 - 3x. The derivative, f'(x), helps us find critical points.
Step 2: Calculate the derivative: f'(x) = 3x^2 - 3.
Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 3 = 0.
Step 4: Solve for x: Factor out 3 to get x^2 - 1 = 0, which gives (x - 1)(x + 1) = 0.
Step 5: Find the critical points: x = 1 and x = -1.
Step 6: To determine the nature of these critical points, find the second derivative of the function: f''(x) = 6x.
Step 7: Evaluate the second derivative at the critical points: f''(1) = 6(1) = 6 (positive) and f''(-1) = 6(-1) = -6 (negative).
Step 8: Interpret the results: Since f''(1) > 0, x = 1 is a local minimum. Since f''(-1) < 0, x = -1 is a local maximum.
Critical Points – Points where the 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 derivative before and after the critical point.
Second Derivative Test – A method used to classify critical points by evaluating the second derivative at those points to determine concavity.
