The y=f (x) function is called increasing on some interval if for any h2> x1 f(x2)> f(x1). If at this f(х2)
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Instruction
1. It is known that for the increasing y=f (x) function its derivative f’ (x)> 0 and according to f’ (x)
2. Example: find intervals of monotony of y=(x^3) / (4-x^2). Decision. Function is defined on all numerical axis, except x =2 and x =-2. To a stern of that it is a nechetna. Really, f (-x)= (-x) ^3) / (4-(-x) ^2) = - (x^3)/(4-x^2) of =f (-x). It means that f(x) is symmetric concerning the beginning of coordinates. Therefore the research behavior of function can be made only for positive values x, and then to complete a negative branch of symmetrically positive. y '= (3 (x^2) (4-x^2) +2x(x^3)) / ((4-x^2) ^2)= (x^2) (12-x^2) / ((4-x^2) ^2).y’ - does not exist at x=2 and x=-2, but at the same time there is no function also.
3. Now it is necessary to find intervals of monotony of function. For this purpose it is necessary to solve inequality: (x^2) (12-x^2) / ((4-x^2) ^2)> 0 or (x^2) (x-2sqrt3) (x+2sqrt3) ((x-2) ^2) ((x+2) ^2)) 0. Use a method of intervals, at the solution of inequality. Then it will turn out (see fig. 1).
4. Further consider behavior of function on monotony intervals, attaching here all data from area of negative values of a numerical axis (owing to symmetry all data there of an obratna including according to the sign).f’ (x)> 0 at – ∞
5. Example 2. To find intervals of increase and decrease of the y=x+lnx/x function. Decision. A function range of definition – x> 0.y '=1+(1-lnx) / (x^2)= (x^2+1-lnx)/(x^2). The sign of a derivative at x> 0 completely is defined by a bracket (x^2+1-lnx). As x^2+1> lnx, y’> 0. Thus, function increases on all the range of definition.
6. Example 3. To find intervals of monotony of the y '=x^4-2x^2-5 function. Decision. y '=4x^3-4x=4x(x^2-1) =4x(x-1) (x+1). Applying a method of intervals (see fig. 2), it is necessary to find intervals of positive and negative values of a derivative. Using a method of intervals, you will be able quickly to define that on x0 intervals the function increases.