The monotony is a determination of behavior of function on a piece of a numerical axis. Function can be monotonously increasing or monotonously decreasing. On the site of monotony the function is continuous.
Instruction
1. If on some numerical interval with growth of an argument the function increases, then on this site the function monotonously increases. The function graph on the site of monotonous increase is directed from below up. If to each smaller value of an argument there corresponds the function size decreasing in comparison with previous, then such function is monotonously decreasing, and its schedule constantly goes down.
2. Monotonous functions have certain properties. For example, the sum of monotonously increasing (decreasing) functions is the increasing (decreasing) function. At multiplication of the increasing function by a constant positive multiplier this function keeps monotonous growth. If the constant multiplier is less than zero, then function from monotonously increasing becomes monotonously decreasing.
3. Borders of intervals of monotonous behavior of function are defined at a function research by means of the first derivative. The physical point of the first derivative function is a speed of change of this function. At the growing function the speed constantly increases, in other words — if the first derivative on some interval is positive, function on this site monotonously increasing. And vice versa — if on a piece of a numerical axis the first derivative of function is less than zero, then this function monotonously decreases in interval borders. If the derivative is equal to zero, then the value of function does not change.
4. For a function research on monotony on the set interval define by the first derivative whether this interval belongs to area of permissible values of an argument. If function on this piece of an axis exists and it is differentiated, find its derivative. Define conditions under which the derivative is more or less than zero. Draw a conclusion on behavior of the studied function. For example, the derivative of linear function is the constant number equal to a multiplier at an argument. At positive value of this multiplier the initial function monotonously increases, at negative — monotonously decreases.