# Functions of Several Variables

• affirmative
In summary, the conversation discusses the question of extracting a value of y using the notation f(x,y) when given a value of x. It is mentioned that in order to evaluate the function, two numbers are needed and the output is a z value, not a y value. It is also explained that with a known z value and x value, it is possible to solve for y using the equation. The conversation concludes with confirming that the explanation makes sense.
affirmative
My question is mostly down to staggering ignorance of basic notation. I have been unable to find a straightforward answer anywhere - so I assume it must be pretty basic.

I have a function: $$f(x,y)=x^3+y^3-3x-12y+20$$

Given that I know a value of x, ie $$x=-1$$, how do I extract a value of y using this notation?

I know when $$f(x)=x^2$$ I can use substitute $$f(x)$$ with $$y$$, but I find myself stuck with what to substitute $$f(x,y)$$ with...

In order to evaluate your f(x, y), you need two numbers. What you get out isn't a y value; it's a z value. To graph the function, you need three dimensions. If you know a z value and an x value, there's the possibility that you can solve for the value you don't know: y. For example, if you know that f(x, y) = 10 and that x = 1, you might be able to solve the equation 10 = 1^3 + y^3 -3(1) - 12y + 20 to get y.

Does that make sense?

Yes, it actually does. Thank you Mark.

## 1. What are the basic concepts of Functions of Several Variables?

The basic concepts of Functions of Several Variables include the domain, range, and graph of the function. The domain refers to the set of all possible inputs or independent variables, while the range is the set of all possible outputs or dependent variables. The graph of a function of several variables is a three-dimensional representation of the function, with the x and y-axis representing the independent variables and the z-axis representing the dependent variable.

## 2. How do you determine the critical points of a Function of Several Variables?

To determine the critical points of a Function of Several Variables, you must first find the partial derivatives of the function with respect to each independent variable. Then, set each partial derivative equal to zero and solve for the values of the independent variables. These values represent the critical points of the function.

## 3. What is the difference between a local and global extremum in a Function of Several Variables?

A local extremum in a Function of Several Variables is a point where the function reaches either a maximum or minimum value in a small region around that point. A global extremum, on the other hand, is the maximum or minimum value of the entire function. Therefore, a global extremum will always be a local extremum, but a local extremum may not necessarily be a global extremum.

## 4. How do you determine the concavity of a Function of Several Variables?

To determine the concavity of a Function of Several Variables, you must first find the second partial derivatives of the function with respect to each independent variable. Then, evaluate these second partial derivatives at a given point. If the value is positive, the function is concave up at that point, and if the value is negative, the function is concave down at that point.

## 5. What is the relationship between partial derivatives and total derivatives in Functions of Several Variables?

The partial derivatives of a function of several variables represent the rate of change of the function with respect to each independent variable. The total derivative, on the other hand, represents the overall rate of change of the function. The total derivative can be calculated by taking the dot product of the gradient vector (a vector containing all the partial derivatives) and the direction vector (a vector that points in the direction of the rate of change).

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