The notion of critical point functions are closely related to the notion of its derivative at that point. Namely, point is called critical if the derivative of a function it does not exist or is zero. Critical points are the internal points of the domain of definition of the function.
To determine the critical points of this function, you must perform several actions: find the domain of the function, calculate derivative, find the domain of definition of the derivative function, find the point treatment of the derivative to zero, to prove the belonging of the points defining the original function.
Example 1Определите critical points of the function y = (x - 3)2·(x-2).
Reinitiate the domain of the function, in this case there are no restrictions: x ∈ (-∞; +∞);Compute the derivative y’. According to the rules of differentiation of product of two functions is: y’ = ((x - 3)2)’·(x - 2) + (x - 3)2·(x - 2)’ = 2·(x - 3)·(x - 2) + (x - 3)2·1. After opening the brackets it turns out the quadratic equation: y’ = 3·x2 – 16·x + 21.
Find the definition of the derivative of the function: x ∈ (-∞; +∞).Solve the equation 3·x2 – 16·x + 21 = 0 to find which x the derivative becomes zero: 3·x2 – 16·x + 21 = 0.
D = 256 – 252 = 4x1 = (16 + 2)/6 = 3; x2 = (16 - 2)/6 = 7/3.So, the derivative becomes zero at values of x equal to 3 and 7/3.
Determine whether found point determining the original function. Since x (-∞; +∞), then both these points are critical.
Example 2Определите critical points of the function y = x2 – 2/x.
Rectiability function definition: x ∈ (-∞; 0) ∪ (0; +∞), since x is in the denominator.Calculate the derivative y’ = 2·x + 2/x2.
The scope of the definition of the derivative function is the same as the original: x ∈ (-∞; 0) ∪ (0; +∞).Solve the equation 2·x + 2/x2 = 0:2·x = -2/x2 → x = -1.
So the derivative vanishes at x = -1. Done necessary but not sufficient condition for criticality. Since x=-1 falls within the interval (-∞; 0) ∪ (0; +∞), this point is critical.
Advice 2 : Interfere whether the critical days weight loss
Many women suffer from premenstrual syndrome, which manifests itself not only pain, but also increased appetite. The result of the critical days can dramatically slow the process of weight loss.
Causes increase appetite during the critical days
The reason for the increase in appetite during the critical days is the change in the overall hormonal levels in the female body. A few days before the onset of menstruation the level of progesterone hormone increases, the body adjusts to the pregnancy and tries to make extra reserves of energy in the form of fat deposits even if a woman is on a diet. Thus, the weight change in the critical days is normal.
When period end, some portion of these reserves goes. However, if you do not control your appetite, gained these days the weight will go away completely. Part of the savings remains in the stomach and sides. To avoid this purchase, a useful habit to weigh yourself before and after your period. The data obtained must write to them then to compare. Making sure that the critical days to slow down the process of weight loss or even promote weight gain, take it calmly, knowing that this is a normal process. You must try to carefully monitor the appetite, the diet and amount of calories eaten during menstruation.
How to eat during menstruation
Try not to eat sweets, pastries and other high calorie products containing "fast" carbohydrates. Their excess is immediately stored in fat. Many women during this period really want to eat chocolate, in this case, you can buy bitter chocolate and treat yourself to a few slices, but not anymore. During menstruation should not drink alcoholic beverages, marinades, pickles, smoked meat, seeds and nuts. Pickles and cured meats in General should be restricted in the diet for 6-8 days before the start of menstruation, because these products increase the water reserves in the body, and this period is characterized by increased accumulation of fluid. To reduce the amount of salt in the diet, add it to the minimum quantity in the finished dish.
It is recommended to consume low-fat dairy products, vegetative food, porridges. Will be useful legumes, boiled potatoes, rice products, which include "slow" carbohydrates. To compensate the loss of iron will help seafood, liver, fish, beef, poultry, eggs, legumes, dried fruits. Will be useful to wheat bran. The natural reaction during menstruation are swelling. Adjust the status light will help diuretic herbs: Basil, dill, parsley, celery. They can be used as a condiment. In the second half of the cycle it is recommended to consume protein products (lean meats and fish, dairy products), and the number of carbohydrates in the diet should be possible to reduce.
Advice 3 : How to determine the critical volume of sales
Economic concept of critical volume of sales corresponds to the position of the enterprise market, in which revenues from the sale of goods is minimal. This situation is called break even point, when the demand for products is falling and profit barely covers the cost. To determine the critical volume of sales, use several methods.
The operating cycle of the enterprise is not limited to its primary activity – production of goods or services. This is a complex organization of labor of a definite structure, including the work of key personnel, managerial staff, managerial staff etc., as well as economists, whose main task is the financial analysis of activity of the enterprise.
The purpose of this analysis is to calculate some values, which in varying degrees affect the size of the final profit. Different volumes of production and sales, the full and average cost of production, indicators of supply and demand, etc. the Main task is to identify this production, which established a strong correlation between cost and profit.
The minimum volume of salesat which revenue covers costs, but does not increase the company's equity capital is called the critical volume of sales. There are three methods of calculation method of this indicator: method of the equations, marginal income, and graphics.
To determine the critical volume of sales on the first method, write down the equation of the form:VP – Sper – Spos = PP = 0, where:VP – revenue from salesand goods;Spar and Spas – variable and fixed costs;PP – profits from salesI.
According to another method, the first term, revenues from sales, present as a product of profit margin from the unit price on the volume of sales, the same applies to variable costs. Fixed costs apply to the entire batch of goods, therefore, this component of shared leave:MD•N – Зпер1•N – Spos = 0.
Express from this equation the value of N and you will get a critical volume of sales:N = Zpos/(MD – Зпер1), where Зпер1 – variable costs per unit of product.
The graphical method involves plotting of graphs of functions. Apply on the coordinate plane two lines: the function of revenue from sales minus both types of costs and the profit function. On the x-axis postpone the amount of products and y-axis and the income of the relevant quantity of the goods expressed in monetary units. The intersection of these lines corresponds to the critical volume of sales, position of break-even.