If , is it always true that
True or False?
If , then
Why some people say it's true: The more you multiply, the bigger it gets, so obviously if , .
Why some people say it's false: It doesn't work for all numbers.
This statement is . In particular, if , then
Proof:
First, we need to understand the graph of the function .
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As shown in the figure above, we have 3 cases:
If , the graph of is monotonically increasing; when , and when .
If , the graph of is monotonically decreasing; when , and when .
If , then .Now consider 3 functions , and where Then we have As and , we have .
Hence, is in the form and , so the graph of is monotonically increasing.When , we have . Since ,
When , we have . Since ,
When , we have . Thus,
Examples
Let and . Find the relationship between and .
We have So, .
Let and . Find the relationship between and .
We have So, .
So, we see that in spite of being greater than , is not always greater than .