Question

Find the range of the functions. $V(x, y)=4 x^{2}+y^{2}$

   Find the range of the functions. 
$V(x, y)=4 x^{2}+y^{2}$

Calculus

Gilbert Strang,… 1st Edition

Chapter 14, Problem 28

Instant Answer

Solved by Expert Amy Jiang

08/06/2019

Step 1

Notice that both x^2 and y^2 are always non-negative (i.e., greater than or equal to 0) since any real number squared is non-negative. Show more…

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Find the range of the functions. $V(x, y)=4 x^{2}+y^{2}$

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Key Concepts

Range of a Function

The range of a function is the complete set of possible output values it can produce when the input variables vary over their entire domain. Determining the range often involves analyzing the structure of the function and identifying the minimum or maximum values it can attain.

Quadratic Forms

Quadratic forms are expressions where the variables are raised only to the second power and typically involve sums of squares multiplied by constants. They are commonly used in multivariable calculus and linear algebra to study the curvature and extremal properties of surfaces and functions.

Positive Definiteness

A quadratic form is positive definite if it always produces positive values (except possibly at the point of minimum where it might yield zero). This concept is crucial when determining the lower bound of a function's range because it guarantees that the function does not output negative values.

Optimization in Multivariable Functions

Finding the range of a multivariable function often involves optimization techniques, where the goal is to find the minimum or maximum values that the function can attain. This involves identifying critical points or leveraging the structure of the function, such as recognizing the nonnegative contributions of squared terms.

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