Question

How do you sketch the graph `y=ln(1/x)` using the first and second derivatives?

Answer 1

`f(x) = ln(1/x)` is monotone and strictly decreasing in its domain and therefore has no local extrema, and it is concave up everywhere.

Explanation:

Using the properties of logarithms we should see that:

`ln(1/x) = -lnx`

If we want to go through the whole sketcing process as an an exercise, we start by noting that `y=ln(1/x)` is defined and continuous for `x in (0,+oo)` and we analyze the limits at the boundaries of the domain:

`lim_(x->0^+) ln(1/x) = lim_(y->+oo) lny = +oo`

`lim_(x->+oo) ln(1/x) = lim_(y->0^+) lny = -oo`

Then we calculate the first and second derivatives:

`d/(dx) ln(1/x) = 1/(1/x) (-1/x^2) =-1/x`

`d^2/(dx^2) ln(1/x) = 1/x^2`

We can see that `f(x)` is monotone and strictly decreasing in its domain and therefore has no local extrema, and that it is concave up everywhere.

graph{ln(1/x) [-10, 10, -5, 5]}