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Chapter 12: Problem 33

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=x^{2}+2 x y-y^{3}$$

Short Answer

Expert verified
Answer: The function \(f(x, y) = x^2 + 2xy - y^3\) is continuous at all points in 鈩澛�.

Step by step solution

01

Analyze the function

The given function is: $$f(x, y) = x^2 +2xy - y^3$$ Now, we will check the continuity of the function by finding its partial derivatives.
02

Find the partial derivatives

The partial derivative of \(f(x, y)\) with respect to \(x\) is: $$\frac{\partial f}{\partial x} = 2x + 2y$$ The partial derivative of \(f(x, y)\) with respect to \(y\) is: $$\frac{\partial f}{\partial y} = 2x - 3y^2$$
03

Check for continuity of partial derivatives

A function is continuous if both its partial derivatives are continuous. In our case, both the partial derivatives are polynomials, and polynomials are continuous for all real numbers. Therefore, both \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) are continuous for all real numbers.
04

Conclude the result

As the partial derivatives of \(f(x, y)\) are continuous for all real numbers, it is safe to conclude that \(f(x, y)\) is continuous at all points in \(\mathbb{R}^2\).

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Most popular questions from this chapter

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