Write the first six terms of the sequence f(n)=n^3+2
1 answer:
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We're given the following function:
![f(x)=log(.75^x)=log[ (\frac{3}{4}) ^{x}]=log( \frac{3^x}{4^x})](https://tex.z-dn.net/?f=f%28x%29%3Dlog%28.75%5Ex%29%3Dlog%5B%20%20%28%5Cfrac%7B3%7D%7B4%7D%29%20%5E%7Bx%7D%5D%3Dlog%28%20%5Cfrac%7B3%5Ex%7D%7B4%5Ex%7D%29%20%20)
In order to see if the function is decreasing we'll take its derivative. If
the function is increasing, if 
the function is decreasing.
We take the derivate:
![\frac{d}{dx}[log( \frac{3^x}{4^x})]= \frac{4^x}{3^x}[ \frac{d}{dx} (\frac{3^x}{4^x})]=\frac{4^x}{3^x} \frac{4^x[ \frac{d}{dx} (3^x)]-3^x[ \frac{d}{dx} (4^x)]}{ 4^2^x}=](https://tex.z-dn.net/?f=%20%5Cfrac%7Bd%7D%7Bdx%7D%5Blog%28%20%5Cfrac%7B3%5Ex%7D%7B4%5Ex%7D%29%5D%3D%20%5Cfrac%7B4%5Ex%7D%7B3%5Ex%7D%5B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28%5Cfrac%7B3%5Ex%7D%7B4%5Ex%7D%29%5D%3D%5Cfrac%7B4%5Ex%7D%7B3%5Ex%7D%20%5Cfrac%7B4%5Ex%5B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%283%5Ex%29%5D-3%5Ex%5B%20%5Cfrac%7Bd%7D%7Bdx%7D%20%284%5Ex%29%5D%7D%7B%204%5E2%5Ex%7D%3D)
![\frac{4^x}{3^x} \frac{4^x[3^xlog(3)]-3^x[4^x[4^xlog(4)]}{ 4^2^x}=log(3)-log(4)\ \textless \ 0](https://tex.z-dn.net/?f=%5Cfrac%7B4%5Ex%7D%7B3%5Ex%7D%20%5Cfrac%7B4%5Ex%5B3%5Exlog%283%29%5D-3%5Ex%5B4%5Ex%5B4%5Exlog%284%29%5D%7D%7B%204%5E2%5Ex%7D%3Dlog%283%29-log%284%29%5C%20%5Ctextless%20%5C%200)
Which implies the function is decreasing.
Another way to answer the problem (although less insightful) you can take any two real numbers
and 
such that 
, then if 
the function is increasing and if 
the function is decreasing. You can verify the function is decreasing with any two numbers in the function's domain.
In order to see if the function is decreasing we'll take its derivative. If
We take the derivate:
Which implies the function is decreasing.
Another way to answer the problem (although less insightful) you can take any two real numbers