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

1 Answer
Andrea S.
Jan 26, 2017

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]}

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