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

1 Answer
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.


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

#f(x) = x - ln(x)#

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''(x) = x^-2# and substituting the x coordinate in, and because #f''(1) > 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'(x) > 1#) or decreasing (#f'(x) < 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.