Question

Show that the function `y=1/(1+tanx)` is decreasing for all values of `x`?

Show that the function `y=1/(1+tanx)` is decreasing for all values of `x`?

Thanks!

Answer 1

Start by finding the derivative.

`y = (tanx + 1)^-1`

By the chain rule, we have

`y' = -sec^2x/(tanx + 1)^2`

We immediately see that `sec^2x` will always be positive no matter the value of `x`. Furthermore, `(tanx + 1)^2` will be positive no matter what value of `x`.

This means that `y'` will be negative wherever the function/derivative is defined, hence the function is uniformly decreasing.

However, with restrictions we see that `x!= (3pi)/4 or (7pi)/4`.

So the function is decreasing but then there are asymptotes, so just keep that in mind. Here is the graph to verify.

Hopefully this helps!