Equations maximize and minimize

Question

Find the maximum and minimum of the equation: $x^2-x+\frac{1}{x^2}+x+1$.

I am randomly trying to substitute values for $x$ but I need a complete method to solve such problems.

Answer 1

This is a method without using calculus.

Let $f\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}$, be the function defined by $f(x)=x^2+\frac{1}{x^2}+1$.

The function $f$ has no maximum because $x^2+\frac{1}{x^2}+1\geq x^2$ and $x^2$ can be as big as we want by getting $x$ bigger.

Consider then the following inequality: $$\left(x-\frac{1}{x}\right)^2\geq 0$$

This inequality is always true because every square of a real number is always nonnegative.

By expanding the left-hand side, we get: \begin{align*} x^2-2+\frac{1}{x^2}&\geq 0\\ x^2+\frac{1}{x^2}&\geq 2\\ x^2+\frac{1}{x^2}+1&\geq 3 \end{align*}

To finally prove that $3$ is the minimum, the points $1$ and $-1$ return $3$ when evaluating the function. So $f(x)\geq 3$ whenever $x\in \mathbb{R}\setminus\{0\}$ with equality when $x=\pm1$.