Give an example of: A function f(x, y) with fx>0 and fy<0 everywhere.

Name: Problem 49, Chapter 14: Calculus Multivariable
Uploaded: 2021-07-14T18:16:32-08:00
Duration: 3 min 16 s
Channel: Leon Druch
Description: Give an example of: A function f(x, y) with fx>0 and fy<0 everywhere.

step 1 All right. We want to try to find a function of x and y where the partial of f with respect to x

Give an example of: A function f(x, y) with fx>0 and fy<0...

Key Concepts

Functions of Several Variables

A function of several variables is one that has more than one independent variable. Instead of mapping a single input to an output, these functions map a vector of inputs to a single output. This concept extends the idea of functions from single-variable calculus to scenarios where multiple factors or dimensions influence the outcome, providing a foundation for studying surfaces and higher-dimensional analogs in mathematics.

Partial Derivatives

Partial derivatives extend the concept of a derivative to functions of several variables by measuring the rate of change of the function with respect to one variable while keeping the others constant. This selective differentiation is crucial in understanding and analyzing the behavior of multivariable functions, particularly in identifying directions of increase or decrease and in solving problems in optimization and differential equations.

Monotonicity in Multivariable Functions

Monotonicity in the context of multivariable functions refers to the behavior of the function with respect to changes in individual variables. When the partial derivative of a function with respect to a variable is positive everywhere, it indicates that the function is strictly increasing along that variable, and when it is negative, the function is strictly decreasing along that variable. This property is integral in ensuring predictable behavior and is widely used in both theoretical analysis and practical applications such as optimization and economic modeling.

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