Exploring Function f(x): Max & Min Values

In summary, the function f(x) = 1/3X^3 - 1/2^2 - 6x + 4 has no global maximum or minimum and is considered "unbounded." It is better to avoid using infinity in this case.
  • #1
physicszman
39
0
consider function f(x) = 1/3X^3 - 1/2^2 - 6x + 4

maximum is positive infinity and minimum is negative infiniti correct?

Thank you!
 
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  • #2
Not if you are looking for local optima. It might be better to say there is no global maximum or minimum, than it is at infinity, to avoid unnecessary use of infinity.
 
  • #3
Originally posted by physicszman
consider function f(x) = 1/3X^3 - 1/2^2 - 6x + 4

maximum is positive infinity and minimum is negative infiniti correct?

Thank you!

I wouldn't put it that way. I would say that the function is "unbounded" and has no global maximum or minimum.
 
  • #4
thanks for the help!
 

Q1. What is the purpose of exploring the function f(x)?

The purpose of exploring the function f(x) is to understand the behavior and characteristics of the function, such as its maximum and minimum values. This can help in analyzing and solving real-world problems that involve the function.

Q2. How do you find the maximum and minimum values of a function f(x)?

To find the maximum and minimum values of a function f(x), you can use calculus techniques such as taking the derivative and setting it equal to zero to find critical points. Then, use the second derivative test to determine if the critical points correspond to a maximum or minimum value.

Q3. Can a function have more than one maximum or minimum value?

Yes, a function can have more than one maximum or minimum value. These are known as local maximum and minimum values, and they occur at critical points where the derivative is equal to zero. However, there can only be one absolute maximum and minimum value for a function.

Q4. How do you determine if a critical point corresponds to a maximum or minimum value?

You can determine if a critical point corresponds to a maximum or minimum value by using the second derivative test. If the second derivative is positive at the critical point, then it is a minimum value. If the second derivative is negative, then it is a maximum value.

Q5. Can a function have a maximum or minimum value at the endpoints of an interval?

Yes, a function can have a maximum or minimum value at the endpoints of an interval. These values are known as absolute maximum and minimum values and can be found by evaluating the function at the endpoints of the interval.

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