# How do you find the inflection points for the function f(x)=x-ln(x)?

Mar 30, 2018

The 'inflection point' is at the coordinate (1,1). With the x coordinate obtained by using the first differential of the function and setting it to equal 0.

#### Explanation:

The 'point of inflection' you mention in your question is most likely referring to the stationary point of said function, as

$f \left(x\right) = x - \ln \left(x\right)$

does not have a 'point of inflection', which basically means the function does not change from a concave to a convex (or vice versa) at any point. However, it does have a stationary point, in which, similar to a point of inflection, means at that very point, the gradient is 0. We can determine the nature of the stationary point by using the second differential, $f ' ' \left(x\right) = {x}^{-} 2$ and substituting the x coordinate in, and because $f ' ' \left(1\right) > 0$ then we know it is in fact a minimum point.

A point of inflection is in fact a stationary point too in the sense that it is also a point on the graph in which the gradient is equal to 0, however, a stationary point may only be called a point of inflection if the function is increasing ( $f ' \left(x\right) > 1$) or decreasing ($f ' \left(x\right) < 1$) on both sides of the stationary point, whereas in a minimum/maximum point, the function will instead transition from a decreasing function to an increasing one, or vice versa.