\documentclass[a4paper,11pt]{article}
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\author{Adam Mackiewicz}
\title{Zadania dodatkowe z matematyki}
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\begin{document}

\maketitle
$ Zadanie 71 f(x, y) = 16 ln(y + 2x) − 8x − y$


$$y+2x>0$$
$$y>-2x$$

$\frac{df}{dx}$=$\frac{16}{y+2x}*2$-8

$\frac{df}{dy}=\frac{16}{y+2x}-2y$

$\frac{df}{dx}$=0$\Rightarrow$ $\frac{32}{y+2x}$-8=0

$\frac{df}{dy}$=0$\Rightarrow$ $\frac{16}{y+2x}-2y=0$

$\frac{32}{y+2x}=8 \Rightarrow 32=8y+16x\Rightarrow 4=y+2x\Rightarrow y=4-2x$

$\frac{16}{4-2x+2x}-2(4-2x)=0\Rightarrow4-8+4x=0\Rightarrow x=1 ,y=2$


$ dla \left\lbrace
\begin{array}{l}
x = 1 \\
y = 2
\end{array}
\right. $
P(1)=(1,2)


punkt podejrzany to P1


$\frac{d^2f}{d^2x}=\frac{-32}{(y+2x)^2}*2=\frac{-64}{(y+2x)^2}$
$\frac{d^2f}{d^2y}=\frac{-16}{(y+2x)^2}-2$
$\frac{d^2f}{dxy}=\frac{-16}{(y+2x)^2*2$
$\frac{d^2f}{dyx}=\frac{-32}{(y+2x)^2*2$
\end{document}