Question

For what values of x is `f(x)=1/(x-3)` concave or convex?

Answer 1

Concave UP on the interval (`-oo, 3`)
Concave DOWN on the interval (`3, oo`)

Explanation:

Take the first derivative of f(x)

`f'(x) = -1/(x-3)^2`

Then second derivative:

`f''(x) = 2/(x-3)^3`

Find the points where `f''(x) = 0` or is undefined.

`f(x)` is undefined when `x=3`

Plug in a number `x>3` and you will see the function will be positive; thus the function is concave up.

Since the multiplicity of the function `(x-3)^3` is an odd function, the values for `x<3` will be negative; thus the function will be concave down.