TITLE: Finding the minimum value of a function without using Calculus
QUESTION [10 upvotes]: Find the minimum value the function $f(x) = x^4 + \frac{1}{x^2}$ when $x \in \Bbb R^*$

My attempt:
Finding the minimum value of this function using calculus is a piece of cake. But since this question appeared on my test when Calculus was not taught to me, there must definitely be a way (Probably by purely using Algebra) to find the minimum value of this function without using Calculus which I am unaware of.
I tried making perfect squares but that got me nowhere. Maybe, I wasn't making the perfect, perfect square :)
Any help would be appreciated.

REPLY [2 votes]: First, the minimum value of $x$ is $b$ such that $f(x) - b$ has a (double) root.  (That is, the amount you must shift the graph of $f$ down so that it meets the $x$-axis once.)  So we are looking for a $b$ so that $f(x) - b = x^4 + \frac{1}{x^2} - b = 0$ has a (double) root.  Observe
$$  x^4 + \frac{1}{x^2} - b = \frac{x^6 - b x^2 + 1}{x^2}  $$
has a root exactly when its numerator does.  So now we just need to know when that cubic in $x^2$ has a double root.
The discriminant of $(x^2)^3 - b(x^2) + 1$ is 
$$  -4(-b)^3 - 27 \cdot (1)^2 = 4b^3 - 27  \text{.}  $$
The discriminant is zero if and only if the polynomial has a double root.  Taking $b = \sqrt[3]{27/4} = \frac{3}{2^{2/3}}$ is the only choice that makes the discriminant zero, so the minimum value of $f$ is this $b$.
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