How can I elegantly prove the inequality $0 < x^4+2x^2-2x+1$ for $x>0$. I have plotted this function in a Sage (an open source and free CAS) and I can see that there is a local min between $0$ and $1$ that lies above the x-axis.
Therefore,I could show that the function is decreasing from $0$ to the local min and then show it is increasing from the local min to infinity and then evaluate the function at the local min and show that it is greater than zero, and hence greater than $0$ for all $x>0$.
- How can I prove this more simply?
A very quick way to show that an expression is non-negative, is to write it as a sum of squares (of real valued expressions). In this case, you can split it into
$$ x^4 + x^2 + (x-1)^2$$
which also shows strict positivity, since not all squares can be simultaneously $0$ here.