Math Problem Statement
solve this
Solution
Let's go step by step through this exercise:
We are given the function , and the goal is to solve the following problems:
1. Prove that can also be written as
We start by simplifying the original function:
We want to express this in the form . Let's rewrite the fraction:
Now, we can split the fraction:
So, we have shown that:
2. Prove that is increasing on the interval
To show that is increasing, we need to check the sign of the derivative .
Recall . Let's calculate the derivative using the quotient rule:
Simplifying the numerator:
Since is always positive for all , is positive for .
Thus, is increasing on the interval .
3a. Prove that admits a minimum and specify it
Since is increasing, the minimum value of occurs at the left endpoint of the interval, which is .
Let's calculate :
Therefore, the minimum value of is .
3b. Prove that admits an upper bound and specify it
Since is increasing and unbounded as , there is no finite upper bound. However, for large values of , approaches 2:
Thus, the function has an asymptote at , which can be considered its upper bound in the limit.
3c. Deduce that is bounded and provide an interval containing
From the previous results, we know that has a minimum of at and approaches 2 as . Therefore, is bounded between and 2.
Thus, the interval containing is:
Would you like further details on any of the steps? Here are some related questions to expand your understanding:
- What is the significance of an asymptote in a function like this?
- Can you explain how the quotient rule for derivatives works?
- How would the behavior of the function change if the denominator were different, say ?
- How can you generalize this process for proving whether functions are increasing or decreasing?
- Why does the function have a minimum, but not a maximum?
Tip: When working with increasing or decreasing functions, the derivative is a crucial tool. If the derivative is positive, the function is increasing, and if negative, it's decreasing!
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Math Problem Analysis
Mathematical Concepts
Functions
Derivative
Increasing functions
Asymptotes
Boundedness
Formulas
Quotient rule for derivatives
Function transformation
Theorems
Derivative test for increasing functions
Limits and asymptotes
Suitable Grade Level
Grades 11-12
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