How to find the maximum value of the expression

Follow finding the maximum value of the function, which in the interval has a finite number of critical points. To do this, calculate its value at all points and at the ends of the segment. From the resulting numbers, choose the highest. The method of finding the greatest value of the expression is used to solve various application tasks.

To do this, run the following steps: translate the task into the language of the function, select x, and through him to Express the required rate as a function of f(x). Using Analytics, find the largest and smallest values of the function at a certain interval.

Use the following examples to find the value of the function. Find the values of the function y=5-square root (4 – x2). Using the definition of the square root, we get 4 - x2 > 0. Solve the quadratic inequality, we get that -2 < x < 2. Divide the resulting interval into two, you get two inequalities -2 < x < 0 and 0 < x < 2.

Erect in the square each of the inequalities, then multiply all three parts by -1, add to them 4. Then enter the auxiliary variable and make the assumption that t = 4 - x2, where 0 < t < 4. A function of y, is the square root of the variable t on the interval is increasing and continuous. Therefore, the greatest value of the function happens at the end of the period.

Perform the inverse change of variables, as a result, you will get the following inequalities: 0 < root of (4 – x2) < 2. Add to all portions 5, before multiplying by -1, you get 3 < 5 - root of (4 – x2). < 5. Thus, the set of values of the function y = 5 - square root (4 – x2) is the interval [3; 5], and the largest value, respectively, 5.

Use the method of application properties of continuous functions to determine the highest value of the expression. In this case, use the numeric values taken by the expression at the specified interval. Among them, there is always a least value m and maximum value M. Between these numbers is a set of values of the function.