16 Questions
What is the condition for a function to have a local maxima?
f(x) < f(c) at x = c
What is a critical number for a function f(x)?
A value of x where f'(x) = 0 or f'(x) does not exist
What is the conclusion if f'(x) changes from positive to negative at x = c?
A local maxima occurs at x = c
What is the purpose of the First Derivative Test?
To determine the local maxima and minima of a function
What is the definition of a critical point?
An ordered pair, where the x-value is the critical number
What happens if f'(x) does not change signs at x = c?
There is no local extrema at x = c
What is the first step in finding the local extrema of a function?
Take the derivative of the function
What is the formula to find the critical points of a function?
F'(x) = 0
What is the purpose of finding the critical points of a function?
To find the local maxima and minima
How do you classify the local extrema of a function?
By finding the second derivative of the function
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
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
What is the purpose of finding the local maxima and minima of a function?
To analyze the behavior of the function
What is the formula for the function given in the problem?
F(x) = 2x^2 + 12x - 18
What is the meaning of the critical points in the context of a function?
The points where the derivative of the function is zero
Why is it important to find the local extrema of a function?
To analyze the behavior of the function
Study Notes
Local Maxima and Minima
- A function has a local maxima if the value of the function at a point
cis 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
cis 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.
Practice Exercises
- Practice exercises can be found on page 163, questions 3, 4, 6ab, and 10ab.
Identify and analyze local maxima and minima of a function using the first derivative test, including critical numbers and points.
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