Minimum of polynom-function

Question

is there a possibility to find/calculate the minimum of the following functions without using a graphical calculator?:

1. $x^{-2} + x^{-1} + x + 1$

2. $x^{-0.5} + x + x^{0.5} + 1$

Answer 1

Solving $x^3-x-2=0$ with Newton method

First we need an interval where the root can be

I made a table

$ \begin{array}{r|r} x & p(x)\\ \hline -2 & -8 \\ -1 & -2 \\ 0 & -2 \\ 1 & -2 \\ 2 & 4 \\ \end{array} $

The solution is in $[1,2]$

Let's guess $x_0=1.5$

Then iteratively we define for $n>0$

$x_{n+1}=x_n-\dfrac{p(x_n)}{p'(x_n)}$

Where $p(x)=x^3-x-2$ and $p'(x)=3x^2-1$

We get the following table

$ \begin{array}{r|r} n & x_n\\ \hline 0 & 1.5 \\ 1 & 1.52174 \\ 2 & 1.52138 \\ 3 & 1.52138 \\ \end{array} $

So the solution is $x\approx 1.52138$

You can try and find the solution to the second

Hope this helps

Answer 2

First use some substitutions:

1. For the first function, $z=x^{-1}$.
2. For the second one, $z=x^{0.5}$ (so $x^{-0.5} = z^{-1}$)

Then you may use derivatives.