Describing Set of Absolute Maxima for a Concave Function

Thread starter garryh22
Start date Apr 9, 2008
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Concave Function

In summary, a concave function is defined as a "downward sloping" or "decreasing" mathematical function where the graph is always below or touching the line between any two points. To find the absolute maxima for a concave function, you must take the derivative and set it equal to 0, then solve for the critical value(s) and plug them back into the original function to find the corresponding y-values. It can only have one absolute maximum due to the graph's decreasing nature. Finding the absolute maxima can help understand the function's behavior and be useful in optimization problems. Other methods for finding the absolute maxima include using the second derivative test.

Apr 9, 2008

garryh22

Homework Statement

Let f(x) be concave, how do you describe its set of absolute maxima

Homework Equations

The Attempt at a Solution

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Apr 9, 2008

HallsofIvy

Science Advisor

Homework Helper

Are you not even capable of reading the rules for this forum? Show what work you have done on this yourself. If nothing else, at least look up and state the definition of "convex" function.

Related to Describing Set of Absolute Maxima for a Concave Function

1. What is a concave function?

A concave function is a mathematical function where the graph is always below or touching the line between any two points on the graph. It is also referred to as a "downward sloping" or "decreasing" function.

2. How do you find the absolute maxima for a concave function?

To find the absolute maxima for a concave function, you must first take the derivative of the function and set it equal to 0. Then, solve for the critical value(s) and plug them back into the original function to find the corresponding y-values. The highest y-value among these critical points is the absolute maximum for the function.

3. Can a concave function have more than one absolute maximum?

No, a concave function can only have one absolute maximum. This is because the graph is always decreasing and can only have one highest point.

4. What is the significance of finding the absolute maxima for a concave function?

Finding the absolute maxima for a concave function can help in understanding the behavior of the function. It can also be useful in optimization problems, where the goal is to find the maximum value of a function.

5. Are there any other methods for finding the absolute maxima of a concave function?

Yes, besides taking the derivative and setting it equal to 0, there is also the second derivative test which involves taking the second derivative of the function and determining its concavity at the critical point. If the second derivative is negative, the critical point is a maximum. If it is positive, the critical point is a minimum.