Finding maximum/minimum values?

  • MHB
  • Thread starter eleventhxhour
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In summary, the maximum value of a function or dataset is the highest point or value it can reach, while the minimum value is the lowest point or value it can reach. These values are important in analyzing data and understanding function behavior. To find them, various methods such as graphing, differentiation, or setting the derivative to zero can be used. They are significant in scientific research as they provide valuable information about systems and processes. A function can have multiple maximum/minimum values, including local and global ones. However, there are limitations to finding these values, such as not always being able to find an exact value and facing difficulties with complex functions or multiple variables.
  • #1
eleventhxhour
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What are the maximum/minimum values for y = 28(1.21)^x on the interval 0 $\le$ x $\le$ 12?

I think that the minimum value might be 28 because y = 28 when x = 0 but I don't know how to find the maximum value. Could someone help/explain? Thanks.
 
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  • #2
An exponential function, whose base $b$ is greater than 1, of the form:

\(\displaystyle y=kb^x\) where $0<k$ will be monotonically increasing on any interval within the domain, which is all reals. Thus, the function will have it's minimum at the left-endpoint and its maximum at the right end-point.
 

FAQ: Finding maximum/minimum values?

What is the definition of maximum/minimum values?

The maximum value of a function or dataset is the highest point or value that it can reach, while the minimum value is the lowest point or value that it can reach. These values are important in analyzing data and understanding the behavior of a function.

How do you find the maximum/minimum values of a function?

To find the maximum/minimum values of a function, you can use various methods such as graphing, differentiation, or setting the derivative to zero. Graphing the function can give you a visual representation of the highest and lowest points. Differentiation can help you find the critical points, which are the points where the derivative is equal to zero. The critical points can then be used to determine the maximum/minimum values.

What is the significance of maximum/minimum values in scientific research?

Maximum/minimum values are significant in scientific research as they can provide valuable information about the behavior and properties of a system or function. They can help in understanding the optimal conditions for a process, identifying key factors that influence a system, and predicting the behavior of a system under different conditions.

Can a function have multiple maximum/minimum values?

Yes, a function can have multiple maximum/minimum values. These are known as local maximum/minimum values, as they occur within a specific interval. A function can also have a global maximum/minimum value, which is the highest/lowest point over the entire domain of the function.

What are the limitations of finding maximum/minimum values?

One limitation of finding maximum/minimum values is that it may not always be possible to find an exact value. In some cases, the maximum/minimum value may be an irrational number, making it difficult to calculate precisely. Additionally, if the function is complex or has multiple variables, it may be challenging to determine the maximum/minimum values accurately.

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