To find the set of values ​​of a function, you first need to know the set of values ​​of the argument, and then, using the properties of inequalities, find the corresponding maximum and minimum values ​​of the function. This leads to the solution of many practical problems.

Instruction

Find the largest value of a function that has a finite number of critical points on a segment. To do this, calculate meaning at all points, as well as at the ends of the segment. From the received numbers, choose the largest. Finding the largest value method expressions used to solve various applied problems.

To do this, perform the following steps: translate the problem into the language of the function, select the parameter x, and express the desired value in terms of it as a function f(x). Using analysis tools, find the largest and smallest values ​​of the function on a certain interval.

Use the following examples to find the value of a function. Find function values ​​y=5-root of (4 – x2). Following the definition of the square root, we get 4 - x2 > 0. Solve the quadratic inequality, as a result you get that -2

Square each of the inequalities, then multiply all three parts by -1, add 4 to them. Then introduce an auxiliary variable and make the assumption that t = 4 - x2, where 0 is the value of the function at the ends of the interval.

Make the reverse change of variables, as a result you will get the following inequality: 0 value, respectively, 5.

Use the method of applying the properties of a continuous function to determine the largest meaning expressions. In this case, use the numeric values ​​that are accepted by the expression on a given segment. Among them there is always the smallest meaning m and the largest meaning M. Between these numbers lies a set of function values.

Instruction

Find the largest , which has a finite number of critical points on the segment. To do this, calculate meaning at all points, as well as at the ends of the segment. From the received, choose the largest. Finding the largest value method expressions for solving various applied problems.

To do this, perform the following steps: translate the problem into the language of the function, select the parameter x, and express the desired value in terms of it as a function f(x). Using analysis tools, find the largest and smallest values ​​of the function on a certain interval.

Count the number of steps needed and think about the order in which they should be done. If this question makes it difficult for you, note that the actions enclosed in brackets are performed first, then division and multiplication; and the subtraction is done last. To make it easier to remember the algorithm of the actions performed, in the expression above each action operator sign (+, -, *, :), with a thin pencil, write down the numbers corresponding to the execution of the actions.

Proceed with the first step, adhering to the established order. Count mentally if the actions are easy to perform verbally. If calculations are required (in a column), record them under the expression, indicating the sequence number of the action.

Clearly track the sequence of actions performed, evaluate what needs to be subtracted from what, what to divide into what, etc. Very often, the answer in the expression turns out to be incorrect due to errors made at this stage.

If you have completed the work suggested to you, I suggest that you check the correctness of its implementation:

No. 1. Solution: a) sin α = -because cos α = 0.6, 1.5 πb) tg (π / 2 + α) = - ctg α = -

#2 Solution:

No. 3. Solution: 6 sinα, because -1 ≤ sinα ≤ 1, then -6 ≤ 6 sinα ≤ 6. So the smallest value of the function is -6, and the largest value of the function is 6.

№ 4. Solution: a) 150 0 = b) 270 0 =

No. 5. Solution: a)

No. 6. Solution: (1 - sin 2 α): (1- cos 2 α) \u003d cos 2 x: sin 2 x \u003d ctg 2 x

I hope that you did not find erroneous solutions on your own or there were very few of them!

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