Question

найти максимум функции

Answer 1

F(x)=х-5/5 + 10/х
f'(x)=1/5-10/x^2=(x^2-50)/5x^2
f'(x)=0
x=5sqrt(2)
x2=-5sqrt(2)
производная при переходе через точку х=5sqrt(2) меняет знак с минуса на плюс
в точке имеется минимум.
в точке х=-5sqrt(2)  максимум

Answer 2

Finding the Maximum of a Function

To find the maximum of a function, you can use various mathematical methods such as differentiation, completing the square, or using critical points. Let's consider an example to illustrate the process.

Suppose we have the function y = x^2 + 4x - 12. To find the maximum of this function, we can start by determining its critical points. The critical points are the points where the derivative of the function is either zero or undefined.

The derivative of the function y = x^2 + 4x - 12 is given by y' = 2x + 4. Setting y' to zero and solving for x gives us the critical point.

y' = 2x + 4 = 0 2x = -4 x = -2

So, the critical point of the function is x = -2. To determine whether this critical point corresponds to a maximum or minimum, we can use the second derivative test. If the second derivative is positive at this point, it corresponds to a minimum, and if it's negative, it corresponds to a maximum.

The second derivative of the function y = x^2 + 4x - 12 is y'' = 2. Since the second derivative is positive, the critical point x = -2 corresponds to a minimum.

Therefore, in this case, the function does not have a maximum value.

This example illustrates the process of finding the maximum of a function using differentiation and critical points. If you have a specific function in mind, feel free to provide it, and I can guide you through the process for that function.