Question

найдите точку максимума функции y= - (x^2+36)/ x Помогите,пожалуйста! Как решать?

Answer 1

y = - (x^2+36)/x
y= -(x+36/x)=-x-36/x
y'=-1+36/x^2
-1+36/x^2=0
1-36/x^2=0
(x^2-36)/x^2=0
(x-6)(x+6)/x^2=0
x=-6;x=6;x=0
   -            +              +             -
-----(-6)------(0)-----(6)----->x => x=6 - точка максимума 
  V             ^             ^           V 

Answer 2

Finding the Maximum Point of the Function y = (x^2 + 36) / x

To find the maximum point of the function y = (x^2 + 36) / x, we need to determine the critical points of the function. Critical points occur where the derivative of the function is equal to zero or undefined.

Let's find the derivative of the function first:

y = (x^2 + 36) / x

To simplify the expression, we can rewrite it as:

y = x + 36 / x

Now, let's find the derivative of y with respect to x:

dy/dx = 1 - 36 / x^2

To find the critical points, we set the derivative equal to zero and solve for x:

1 - 36 / x^2 = 0

Multiplying both sides by x^2:

x^2 - 36 = 0

Factoring the equation:

(x - 6)(x + 6) = 0

Setting each factor equal to zero:

x - 6 = 0 or x + 6 = 0

Solving for x:

x = 6 or x = -6

Now that we have the critical points, we need to determine whether they correspond to a maximum or minimum. We can do this by analyzing the second derivative of the function.

Taking the second derivative of y with respect to x:

d^2y/dx^2 = 72 / x^3

Since the second derivative is positive for all values of x, we can conclude that the critical points correspond to a minimum. Therefore, there is no maximum point for the function y = (x^2 + 36) / x.

I hope this helps! Let me know if you have any further questions.