Explain which number is greatest: x^(x+1) or (x + 1)^x when x > 1.
why not just pick a number?
What if x=2?
2^(2+1) = 2^3 = 8
(2+1)^2 = 3^2 = 9.
Got it, thanks! Sorry, I think I'm overthinking things sometimes. I am grateful for the help!
(I also know that it changes when you go past 2, technically too, forgot to write that)
To determine which number is greatest, we need to compare the expressions x^(x+1) and (x + 1)^x when x is greater than 1.
Let's first simplify the expressions:
x^(x+1) can be re-written as x * x^x, and
(x + 1)^x can be re-written as (x + 1) * (x + 1)^(x-1).
Now, let's compare these two expressions:
To compare x * x^x and (x + 1) * (x + 1)^(x-1), we need to consider the growth rate of each term as x increases.
For x * x^x, as x increases, the terms x * x^x also increase.
For (x + 1) * (x + 1)^(x-1), as x increases, the terms (x + 1) * (x + 1)^(x-1) also increase.
Now, let's take the ratio of these two expressions and compare the limits as x approaches infinity:
lim (x->∞) [x * x^x / (x + 1) * (x + 1)^(x-1)]
We can simplify this further:
lim (x->∞) [(x/x+1) * (x^x / (x + 1)^(x-1))]
At this point, we can use the concept of limits and perform some algebraic manipulations to find the limit.
By taking the limit as x approaches infinity, the ratio approaches 1. Therefore, both expressions x * x^x and (x + 1) * (x + 1)^(x-1) have essentially the same growth rate as x increases.
In conclusion, for x > 1, there is no clear maximum value between x^(x+1) and (x + 1)^x.