0
15

{frequent}
{searched}
{frequent}
{searched}
cd
{frequent}
{searched}

How to find the value of the function $f(x,y)$ where $f(x,y) = x^3 − 5xy^2$

The question is as follows:
Let $f(x,y) = x^3 − 5xy^2$ and $g(x,y) = y^2 − x^2$ be real-valued functions and suppose that $g(x,y) \ge 0$ for all $x,y ∈ \mathbb R$. The question asks us to find the value of the function $f(x,y)$ where $f(x,y) = x^3 − 5xy^2$
I have set up the question as $$\int_{x=a}^{b} f(x,y) dx \;.$$ and I have used the basic formula: $$\int_{\frac{b}{a}}^1 \frac{1}{t^2} dt = \ln \left | \frac{b}{a} \right | \;.$$
I have not tried to find the solution to the integral as I am not too sure how to go about this. Any help would be appreciated.

A:

You can directly try to find the possible values of the function $f(x,y)$ by means of the implicit function theorem; the set of critical points of $f(x,y)$ being defined by
$$f_x(x,y)=x^3-5xy^2=0\implies x=\frac{5y^2}{2x}\implies y=\pm\sqrt\frac{2x}{5}$$
where the positive sign corresponds to the solutions of the system, and the negative sign corresponds to the solutions of the system if they exist.
As we have a quadratic equation in $x$, the function has two roots, so the set is finite; hence the function is non-positive everywhere, as it admits only zero as a global minimum.

If you