Question

You are told that there is a function \(f\) whose partial derivatives are \({f_x}(x,y) = x + 4y\) and \({f_y}(x,y) = 3x - y\). Should you believe it?

Short answer

A partial differential equation (PDE) is a mathematical equation that establishes relationships between the partial derivatives of a multivariable function.

Step-by-step solution

Partial derivative

Let \({f_x}(x,y) = x + 4y\)

\({f_{xy}}(x,y) = 4\)

\({f_y}(x,y) = 3x - y\)

\({f_{yx}}(x,y) = 3\)

Step 2:

Since \({f_{yx}}(x,y)\) and \({f_{xy}}(x,y)\) both are continuous everywhere.

But \({f_{yx}}(x,y) \ne {f_{xy}}(x,y)\)

So, this is the contradiction with Clariraut’s theorem.

Hence, there is no function \(f(x,y)\) for which \({f_x}(x,y) = x + 4y\), and \({f_y}(x,y) = 3x - y\).

\(\therefore \)I shouldn’t believe it.