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broadcomreader 0.8.0.0Q:

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.

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John Charles Fernald (February 17, 1889 – May 13, 1958) was an American composer and teacher.

Biography
Fernald was born in Southampton, New York, of Quaker parents. His first exposure to musical training was with his father, who had been trained by Dr. Henry E. Haight. John was one of seven children of John and Jennie Fernald. His father was head of the music department at Southampton College, where he taught the violin, viola, flute, and oboe. John Fernald attended that college, graduating in 1910. He then enrolled in the Eastman School of Music, where he received both his bachelor’s (1913) and master’s (1914) degrees. He continued his studies at Princeton University (1914–15), and at Yale University (1915–16), where he studied with Vincent d’Indy and George Whitefield Chadwick. After graduation, Fernald spent a year teaching music at the H
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