Calculus: Minimums and Maximums of Functions

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Match the following definitions with the corresponding type of critical point:

A point where the function changes from decreasing to increasing = Local minimum A point where the function changes from increasing to decreasing = Local maximum A point where the derivative is equal to 0 or undefined = Critical point A point where the function changes from decreasing to increasing or increasing to decreasing = Saddle point

Match the following statements with the corresponding test:

If f'(a) changes sign from negative to positive at x=a, then f(x) has a local minimum at x=a = First Derivative Test If f''(a) > 0, then f(x) has a local minimum at x=a = Second Derivative Test If f'(a) does not change sign at x=a, then f(x) has a saddle point at x=a = First Derivative Test If f'(a) changes sign from positive to negative at x=a, then f(x) has a local maximum at x=a = First Derivative Test

Match the following characteristics with the corresponding type of extrema:

The smallest value of f(x) over its entire domain = Global minimum The largest value of f(x) over its entire domain = Global maximum A point where the function changes from decreasing to increasing = Local minimum A point where the function changes from increasing to decreasing = Local maximum

Match the following applications with the corresponding field:

Finding the minimum or maximum of a function to solve real-world problems = Optimization Describing the behavior of physical systems = Physics and engineering Determining the critical points of a function = Calculus Graphing functions to visualize their behavior = Graph theory Signup and view all the answers

Match the following equations with the corresponding type of extrema:

f''(a) > 0 = Local minimum f''(a) < 0 = Local maximum f'(a) = 0 = Critical point f'(a) changes sign from negative to positive at x=a = Local minimum Signup and view all the answers

Match the following statements with the corresponding type of extrema:

A point where the derivative is equal to 0 or undefined = Critical point A point where the function changes from decreasing to increasing = Local minimum The smallest value of f(x) over its entire domain = Global minimum The largest value of f(x) over its entire domain = Global maximum Signup and view all the answers

Match the following characteristics with the corresponding type of extrema:

A point where the function changes from increasing to decreasing = Local maximum A point where the function changes from decreasing to increasing = Local minimum The largest value of f(x) over its entire domain = Global maximum The smallest value of f(x) over its entire domain = Global minimum Signup and view all the answers

Match the following tests with the corresponding application:

First Derivative Test = Determining the type of critical point Second Derivative Test = Determining the type of local extrema Global minimum and maximum = Optimization problems Critical points = Calculus Signup and view all the answers

Match the following definitions with the corresponding concept:

Match the following statements with the corresponding type of test:

If f'(a) changes sign from negative to positive at x=a, then f(x) has a local minimum at x=a = First Derivative Test If f''(a) > 0, then f(x) has a local minimum at x=a = Second Derivative Test If f'(a) does not change sign at x=a, then f(x) has a saddle point at x=a = First Derivative Test If f'(a) changes sign from positive to negative at x=a, then f(x) has a local maximum at x=a = First Derivative Test Signup and view all the answers

Study Notes

Minimums and Maximums of Functions

Critical Points

A critical point of a function f(x) is a point where the derivative f'(x) is equal to 0 or undefined.
Critical points can be local minimums, local maximums, or saddle points.

Local Minimums and Maximums

A local minimum of a function f(x) is a point where the function changes from decreasing to increasing.
A local maximum of a function f(x) is a point where the function changes from increasing to decreasing.
A local minimum or maximum is also a critical point.

First Derivative Test

If f'(a) changes sign from negative to positive at x=a, then f(x) has a local minimum at x=a.
If f'(a) changes sign from positive to negative at x=a, then f(x) has a local maximum at x=a.
If f'(a) does not change sign at x=a, then f(x) has a saddle point at x=a.

Second Derivative Test

If f''(a) > 0, then f(x) has a local minimum at x=a.
If f''(a) < 0, then f(x) has a local maximum at x=a.
If f''(a) = 0, then the test is inconclusive.

Global Minimums and Maximums

A global minimum of a function f(x) is the smallest value of f(x) over its entire domain.
A global maximum of a function f(x) is the largest value of f(x) over its entire domain.
Global minimums and maximums can occur at critical points or at the endpoints of the domain.

Applications

Optimization problems: finding the minimum or maximum of a function can be used to solve real-world problems, such as minimizing cost or maximizing profit.
Physics and engineering: minimums and maximums are used to describe the behavior of physical systems, such as the trajectory of a projectile.