The second derivative test is helpful. At a critical number c for which f'(c) = 0, if f''(c) > 0, (c, f(c)) is a local minimum point; if f''(c) < 0, (c, f(c)) is a local maximum point.crisalyn said:hi!
how determine whether that point is the maximum or the minimum?
Maxima-minima problems are mathematical optimization problems that involve finding the maximum or minimum value of a function within a given range of inputs. These types of problems are commonly encountered in fields such as economics, physics, and engineering.
To determine if a critical point is a maximum or minimum, you can use the first or second derivative test. The first derivative test involves evaluating the first derivative of the function at the critical point. If the first derivative is positive, the critical point is a minimum. If the first derivative is negative, the critical point is a maximum. The second derivative test involves evaluating the second derivative of the function at the critical point. If the second derivative is positive, the critical point is a minimum. If the second derivative is negative, the critical point is a maximum.
To solve a maxima-minima problem, you can follow these steps:
1. Identify the function to be optimized.
2. Determine the domain of the function.
3. Find the critical points of the function by setting the first derivative equal to 0 and solving for the input values.
4. Use the first or second derivative test to determine if the critical points are maximum or minimum.
5. Evaluate the function at the critical points and at the endpoints of the domain.
6. Compare the values to determine the maximum or minimum value of the function.
An absolute maximum or minimum is the highest or lowest point of a function within its entire domain, while a relative maximum or minimum is the highest or lowest point within a specific interval or range. In other words, a relative maximum or minimum is a local extremum, while an absolute maximum or minimum is a global extremum.
Maxima-minima problems can be applied in various real-life situations, such as determining the maximum profit a company can make, finding the minimum cost for a construction project, or optimizing the design of a product to achieve the maximum performance. These problems can also be used in fields like medicine, where doctors may need to find the maximum or minimum dosage of a medication for a patient.