Here are a number of step-by-step function optimization issues that have been resolved.
To understand the procedure for solving these problems, you need to understand the applications separately from the derivative ones, such as how to calculate maximums and minimums in a function.
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How to solve the problems of optimization of functions of two variables
In order to solve the problems of function optimization, we must first of all propose the equation indicated by the problem statement. If we have two variables, we need two equations that relate to those two variables.
Then we need an equation that depends on only one variable. To do this, we must clear a variable in one of the equations and replace the expression we have left in the other, just as we do to solve a system of two equations by the substitution method.
What we have left is a function that will depend on a variable from which we will have to find its maximum or minimum (depending on what the problem asks us), applying derivatives. This will be one of the solutions.
We can only substitute the value obtained in the expression where we clear the other variable to obtain the other solution.
To make it clear to you, let’s take a step-by-step look at it with some determined exercises.
Optimized function problems solved
Break down the number 16 into two positive summands so that your product is maximum.
The problem statement is asking for two numbers, x and y, so that when you add them together the result is 16. From here we get the first equation.
And that the product of these two numbers is maximum, that is, that by multiplying x by y, we have the greatest possible result. From here we get the second equation and as we do not know the result of x per y, we call the product P:
We are going to convert this equation so that it depends on only one unknown factor, for example x.
From the first equation, we cleared the y:
And we substitute this expression in the second equation, in the place of the y, and we operate:
We have left a function that only depends on the x, so we can write it like this:
This function represents the product of two numbers, so if we find the maximum of the function, we will be finding the solution to the problem.
To find the maximum of the function, we need the first derivative, so we calculate it:
We set the function to 0:
And we solve the equation:
We have obtained a value of x which is a possible maximum.
To check if it is a maximum, we calculate the second derivative:
And we check its sign for the value of the second derivative in x=8
Therefore, the first number asking us for the problem is:
And to calculate the second, let’s use the expression where we clear the y:
Substituting the y for the 8:
Which results in the second number:
The sum of the two numbers gives 16 and the product is the maximum possible:
Find a number so that when you subtract its square the difference is maximum and greater than zero.
From the statement we get the following function:
In this function we have to look for the maximum.
To do this, we calculate the first derivative:
We set the function to zero:
And we solve the equation:
Where we get a possible maximum.
To check if it is maximum, we get the second derivative of the function:
And we calculate the value of the second derivative is that point, which we see is less than zero, then at that point there is a maximum:
And so the number we’re looking for is:
A sheet of paper should contain 18 cm² of printed text. The top and bottom margins should be 2 cm high each and the sides 1 cm. what dimensions should the sheet of paper have so that paper consumption is minimal?