How can you prove that for all x, where x is a positive real number, $1+(1/(x^4)) ≥(1/x)+(1/(x^3))?$

Question

I am trying to find a way to prove this, so far I have only gotten to

$x^4 + 1 ≥ x^3 + x$

Could you please help me?

Answer 1

It is enough to prove that

$$ x^4-x^3-x+1\geq 0. $$

We have

$$x^4-x^3-x+1=x^3(x-1)-(x-1)=(x-1)(x^3-1)=(x-1)^2(x^2+x+1).$$

Both factors are nonnegative for any positive real $x$ so the relation above holds true.

Answer 2

Subtract $2x^2$ from your last inequality, then $$ \iff (x^2-1)^2\ge x(x-1)^2\iff (x+1)^2\ge x\iff (2x+1)^2+3\ge0 $$ and the last is certainly true.