Find a bound for the function

Question

Consider $x, y$ real

Find a upper bound function for

$$\bigg | \frac{x^3 y^4}{x^6 + y^6} \bigg |$$

I am trying to find an upper bound for this function, but have been unable to do so thus far.

How can I manipulate this using the triangle inequality?

Answer 1

HINT:

Using the AM-GM inequality, we have $x^6+y^6\ge 2\sqrt{x^6y^6}=2|x|^3|y|^3$, which provides an upper bound of $|y|$.

Answer 2

Hint: you could use $\left(x^3-y^3\right)^2 \ge 0 \implies x^6+y^6 \ge 2x^3y^3$, so: $$\left| \frac{x^3 y^4}{x^6 + y^6}\right| \le \left| \frac{x^3 y^4}{2x^3y^3}\right| = \ldots$$