A complete study of the function and the construction of its graph (9x) / (9 + x ^ 2).

1) 9 + x ^ 2 <> 0 – holds for any x, hence the function is defined on the entire set of rational numbers.

Range of values from minus infinity to plus infinity.

2) Find the derivative:

y ‘= (9x / (9 + x ^ 2)’ = (9 (x + x ^ 2) – 9x * 2x)) / (9 + x ^ 2) ^ 2.

We equate it at zero:

9x – 9x ^ 2 = 0;

x * (1 – x) = 0;

x1 = 0; x2 = 1. – extreme points;

3) y ‘<0 on the intervals from minus infinity to 0 and from 1 to infinity – the function decreases, y’> 0 on the interval (0; 1) – increases.

4) x0 = 0 is a minimum point, x0 = 1 is a maximum point.