Question

Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval: f(x) = cot(x), (0,pi)

Answer 1

cot x is just cos x / sin x. on the interval (0, pi), cosine decreases, and sin increases and decreases. so cos(x) is strictly decreasing but sin(x) increases, then starts to decrease, so the ratio between the two isn't very clear.

take a derivative to analyze it further

d/dx cot x = - csc^2 (x)

-csc^2 (x) is the same as -1/sin^2(x)

because sin^2(x) is always positive, that makes -1/sin^2(x) always negative

of course we are talking about the domain (0, pi) since d/dx isn't defined at 0 or at pi.

so if the derivative is always negative on the interval, what does that tell us about the original function ?

Answer 2

it is always decreasing, so it is strictly monotonic.

Answer 3

Makes sense now, Thank you so much!

Answer 4

yes you got it