First Derivative Test (Local Extrema)

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the condition for a function to have a local maxima?

f(x) > f(c) at x = c

f(x) < f(c) at x = c (correct)

f(x) != f(c) at x = c

f(x) = f(c) at x = c

What is a critical number for a function f(x)?

A value of x where f(x) is not defined

A value of x where f'(x) = 0 or f'(x) does not exist (correct)

A value of x where f'(x) > 0

A value of x where f(x) = 0

What is the conclusion if f'(x) changes from positive to negative at x = c?

There is no local extrema at x = c

The function is not continuous at x = c

A local minima occurs at x = c

A local maxima occurs at x = c (correct)

What is the purpose of the First Derivative Test?

To determine the local maxima and minima of a function Signup and view all the answers

What is the definition of a critical point?

An ordered pair, where the x-value is the critical number Signup and view all the answers

What happens if f'(x) does not change signs at x = c?

There is no local extrema at x = c Signup and view all the answers

What is the first step in finding the local extrema of a function?

Take the derivative of the function Signup and view all the answers

What is the formula to find the critical points of a function?

F'(x) = 0 Signup and view all the answers

What is the purpose of finding the critical points of a function?

To find the local maxima and minima Signup and view all the answers

How do you classify the local extrema of a function?

By finding the second derivative of the function Signup and view all the answers

What is the relationship between the critical points and the local extrema of a function?

The critical points are used to find the local maxima and minima Signup and view all the answers

How do you find the local maxima and minima of a function?

By finding the second derivative of the function and evaluating it at the critical points Signup and view all the answers

What is the purpose of finding the local maxima and minima of a function?

To analyze the behavior of the function Signup and view all the answers

What is the formula for the function given in the problem?

F(x) = 2x^2 + 12x - 18 Signup and view all the answers

What is the meaning of the critical points in the context of a function?

The points where the derivative of the function is zero Signup and view all the answers

Why is it important to find the local extrema of a function?

To analyze the behavior of the function Signup and view all the answers

Study Notes

Local Maxima and Minima

A function has a local maxima if the value of the function at a point c is greater than the values of the function at nearby points.
A function has a local minima if the value of the function at a point c is less than the values of the function at nearby points.

Critical Numbers and Critical Points

A critical number is a point where the derivative of a function is equal to zero or does not exist.
A critical point is an ordered pair, where the x-value is the critical number and the y-value is the function value at that point.

The First Derivative Test

The test is used to determine if a critical point is a local maximum or minimum.
If the derivative changes from positive to negative at a critical point, then it is a local maximum.
If the derivative changes from negative to positive at a critical point, then it is a local minimum.
If the derivative does not change signs at a critical point, then it is not a local maximum or minimum.

Examples

To find the critical points, set the derivative equal to zero and solve for x.
To determine the type of critical point, analyze the sign of the derivative on either side of the critical point.