Do a thorough investigation of the function y = x ^ 3-6x ^ 2.

The solution of the problem:
Scope: the entire numeric axis.
1) Find the points of intersection with the X coordinate axis (that is, with the abscissa axis). To do this, we equate the variable y to zero: y = 0.
x ^ 3 – 6 * x ^ 2 = 0.
Pull x ^ 2 out of the bracket:
x ^ 2 * (x – 6) = 0;
The product is zero when one of the factors is zero:
x ^ 2 = 0 or x – 6 = 0;
x = 0 or x = 6.
Points (0; 0), (6; 0) – points of intersection with the 0X axis.
2) The graph of the function crosses the ordinate axis when x = 0.
y = 0 ^ 3 – 6 * 0 ^ 2;
y = 0.
Point (0; 0) is the point of intersection with the 0Y axis.
3) To find the extrema of the function, it is necessary to find the derivative and equate it to zero.
y ‘= (x ^ 3 – 6 * x ^ 2)’ = 0;
3 * x ^ 2 – 12 * x = 0;
x = 0 or x = 4.
y (0) = 0.
y (4) = – 32.
(0; 0) and (4; – 32) are the extrema of the function.
4) Check for evenness (oddness):
y (- x) = (- x) ^ 3 – 6 * (- x) ^ 2 = – x ^ 3 – 6 * x.
Therefore, the function is neither even nor odd.
5) The function decreases on the intervals: (- oo; 0] and [4; + oo).
Decreases by [0; 4].
6) Inflection points:
y ” = 0;
(x ^ 3 – 6 * x ^ 2) ” = 0;
6 * x – 12 = 0;
x = 2 – inflection point.
The function is concave at [2; + oo) and is curved on (- oo; 2].