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Questions and Answers
Match the following definitions with the corresponding type of critical point:
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:
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:
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:
Match the following applications with the corresponding field:
Match the following equations with the corresponding type of extrema:
Match the following equations with the corresponding type of extrema:
Match the following statements with the corresponding type of extrema:
Match the following statements with the corresponding type of extrema:
Match the following characteristics with the corresponding type of extrema:
Match the following characteristics with the corresponding type of extrema:
Match the following tests with the corresponding application:
Match the following tests with the corresponding application:
Match the following definitions with the corresponding concept:
Match the following definitions with the corresponding concept:
Match the following statements with the corresponding type of test:
Match the following statements with the corresponding type of test:
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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.
Critical Points
- Critical points of a function f(x) occur 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.
- Local minimums and maximums are also critical points.
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 use minimum or maximum values to solve real-world problems, such as minimizing cost or maximizing profit.
- Minimums and maximums are used in physics and engineering to describe the behavior of physical systems, such as the trajectory of a projectile.
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