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

Question

Evelyn Agard · Accepted Answer

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 stationary point of the function in question is most likely what you mean when you refer to the "point of inflection" in your question. #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.

Elijah Abram · Answer

undefined To find the inflection points of the function $ f(x) = x - \ln(x) $, you need to follow these steps:

1. Calculate the second derivative of the function $ f(x) $.
2. Set the second derivative equal to zero and solve for $ x $.
3. Determine the corresponding $ y $ values for the found $ x $ values.
4. Verify whether the concavity changes at these points.

Let's proceed with the steps:

1. The first derivative of $ f(x) $ is $ f'(x) = 1 - \frac{1}{x} $.
2. The second derivative of $ f(x) $ is $ f''(x) = \frac{1}{x^2} $.
3. Setting $ f''(x) = 0 $ gives $ \frac{1}{x^2} = 0 $, which has no real solutions because the denominator cannot be zero.
4. Thus, there are no inflection points for the function $ f(x) = x - \ln(x) $.