Question

Find the points of inflection of the curve below. Round the answers to three decimal places. (Hint: All three are in a straight line.)
y= (1+x)/(1+x^2)

Answer 1

Geometrically an inflection point is a point on the curve where the concavity changes. Mathematically, they are always found at points where f"(x)=0 and where the value of f"(x) changes sign. However, f"(x)=0 is a necessary but not sufficient condition for the existence of an inflection point. It is possible that f"(c)=0 but c is not an inflection point. An example is f(x)=x^4 at x=0.

It is therefore important to analyze the function for asymptotes and discontinuities before proceeding to avoid surprises. Verify that the solutions to f"(x)=0 are indeed inflection points.

Given f(x) = (1+x)/(1+x^2)

First, examine the domain and range of the function: both domain and range are on the interval [-∞,+∞], so there are no vertical asymptotes.

As x tends towards ±∞, f(x)-> ±0. So the function approaches ±0 at ±∞.


Now examine the first derivative and its zeroes which are at -√2 - 1 and √-1 respectively. These are the possible extremums.

Note that the common denominator is always positive for all values of x, therefore the solution of f'(x)=0 can be found by equating the numerator (a quadratic) to zero.


It's now time to look at the second derivative and its zeroes, which are found at x=-√3-2, x=√3-2 and x=1. Verify that that these are inflection points by checking that f"(c-) and f"(c+) have opposing signs.