Rearrange The Terms First

Relevant wiki: Applying the Arithmetic Mean Geometric Mean Inequality

We cannot have $x$ as negative and $y$ as positive since $-54xy$ will be positive, and so will the other terms, but we can easily see that the expression can attain a negative value, so we can have either $x,y$ as both positive or both negative. Nevertheless, AM-GM is still applicable.

By AM-GM,

$\frac{x^6+y^6+27+27+27+27}{6} \geq \sqrt[6]{x^6\cdot y^6 \cdot 3^{12}}$ .

This gives us $\frac{x^6+y^6+27+27+27+27}{6} \geq 9xy$ or $x^6+y^6 \geq 54xy-108$

Subtracting $54xy$ to both sides gives us $x^6+y^6 -54xy \geq -108$

So, our answer is $\boxed{-108}$ .

Let me consider: $x^6 + y^6 = (x^2)^3 +( y^2)^3$

let $x^2= a$ and $y^2 = b$

Clearly $a,b>0$ and hence I can apply AM GM inequality on them

$a^3 + b^3 \ge 2 \sqrt{a^3b^3} = 2(xy)^3$ or $2(-xy)^3$

Now there arise two conditions :

1. When $xy<0$ ( no minimum can be achieved in this case , u may confirm it by differentiation)
2. When $xy >0$

Now let $xy = t$ , using this substitution the equation can be written as

$x^6 + y^6 -54xy \ge 2t^3 - 54t$

Using calculus u may calculate the minimum value of $t^3 - 54t$ which will turn out to be $-108$

Using AM-GM inequality we have:

$x^6+y^6-54xy\geq 2|xy|^3-54xy$ .

Let $a=xy$ . Thus we have to minimise the one variable function $f(a)=2|a|^3-54a$ . This can be done very easily using calculus, but I would like to apply AM-GM again, using an approach similar to @Manuel Kahayon .

Consider first the case $a>0$ . I need to add e subtract some (two in this case because we have $a$ to the power of $3$ ) positive terms in order to apply the inequality and I would like to get $+54a$ when these terms are multiplied together. So I can rewrite the function:

$2a^3+b_1+b_2-b_1-b_2-54a\leq 3(2a^3b_1b_2)^{\frac{1}{3}}-b_1-b_2-54a$ .

Now if $b_1=b_2=54$ we have $3(2*54^2a^3)^{\frac{1}{3}}=54a$ and so we have:

$2a^3+54+54-108-54a\leq -108$

When $a=3>0$ we have affectively $f(3)=-108$ so $-108$ is a minimum when $a>0$ .

Moreover, when $a\leq 0$ we have $f(a)\geq 0$ and so $-108$ is a global minimum of $f(a)$ .

Assume x = y, they have a similar relationship, and simple is more efficient. So we have 2x^6 - 54x^2. Take the derivative. You get 12x^5 - 108x. You want this to equal 0. So we have 12x^5 = 108x. 12x^4 = 108. x^4 = 9. x^2 = 3. Plug in x^2 = 3 into 2x^6 - 54x^2. 54 - 162 = -108.