How create these boxes style?

Question

I want to create these theorem boxes

Answer 1

1. set attach boxed title to top left to get a boxed title on the top left.
2. rule above the title and the custom title box is drawn in boxed title style={overlay={...}}
3. rule above the frame and shade below the title is drawn in overlay unbroken={...}
4. left rule is drawn by borderline west=...
5. shadow is controled by drop fuzzy shadow

\documentclass{article}
\usepackage{tcolorbox}
\usepackage{lipsum}
\tcbuselibrary{skins}
\usetikzlibrary{shadings}
\newcounter{example}
\colorlet{colexam}{red!75!black}
\tcbset{
  base/.style={
    empty,
    frame engine=path,
    colframe=yellow!10,
    sharp corners,
    title={Example \thetcbcounter},
    attach boxed title to top left={yshift*=-\tcboxedtitleheight},
    boxed title style={size=minimal, top=4pt, left=4pt},
    coltitle=colexam,fonttitle=\large\bfseries\sffamily,
  }
}
\newtcolorbox[use counter=example]{myexamplea}{%
  base,
  boxed title style={overlay={
    \draw[colexam,line width=3pt,] (frame.north west)--(frame.north east);
  }},
  colback=colexam,
  overlay unbroken={
    \draw[colexam] ([yshift=-1.5pt]title.north east)--([xshift=-0.5pt, yshift=-1.5pt]title.north-|frame.east);
  },
}
\newtcolorbox[use counter=example]{myexampleb}{%
  base,
  drop fuzzy shadow,
  borderline west={3pt}{-3pt}{colexam},
}
\newtcolorbox[use counter=example]{myexamplec}{%
  base,
  drop fuzzy shadow,
  coltitle=black,
  borderline west={3pt}{-3pt}{teal!50},
  attach boxed title to top left={xshift=-3mm, yshift*=-\tcboxedtitleheight/2},
  boxed title style={right=3pt, bottom=3pt, overlay={
    \draw[draw=teal!70, fill=teal!70, line join=round]
      (frame.south west) -- (frame.north west) -- (frame.north east) --
      (frame.south east) -- ++(-2pt, 0) -- ++(-2pt, -4pt) --
      ++(-2pt, 4pt) -- cycle;
  }},
  overlay unbroken={
    \scoped \shade[left color=teal!10!black, right color=teal]
    ([yshift=-0.2pt]title.south west) -- ([xshift=-1.5pt, yshift=-0.2pt]title.south-|frame.west) -- ++(0, -6pt) -- cycle;
  },
}
\begin{document}
\begin{myexamplea}
  \lipsum[1][1-3]
\end{myexamplea}
\begin{myexampleb}
  \lipsum[1][1-3]
\end{myexampleb}
\begin{myexamplec}
  \lipsum[1][1-3]
\end{myexamplec}
\end{document}

Answer 2

Straight from the book -- http://mirror.iopb.res.in/tex-archive/macros/latex/contrib/tcolorbox/tcolorbox.pdf

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{varioref}
\usepackage{tcolorbox}
\tcbuselibrary{skins}
\usepackage{cleveref}
\newtcolorbox{YetAnotherTheorem}[1]%
  {enhanced,arc=0mm,outer arc=0mm,
  boxrule=0mm,toprule=1mm,bottomrule=1mm,left=1mm,right=1mm,
  titlerule=0mm,toptitle=0mm,bottomtitle=0mm,top=0mm,
  colframe=red!50!black,colback=red!5!white,coltitle=red!50!black,
  colbacktitle=yellow!50!white,colback=red!5!white,
  title=#1,
  fonttitle=\bfseries\sffamily\normalsize,fontupper=\normalsize\itshape,
  }
\begin{document}
\begin{YetAnotherTheorem}{Mittelwertsatz f"{u}r $\mathbf{n}$ Variable}
  $f\in C^{1}(D,\mathbb{R})$. Dann gibt es auf jeder Strecke
  $[x_0,x]\subset D$ einen Punkt $\xi\in[x_0,x]$, so dass gilt
    \begin{equation}
    f(x)-f(x_0) = \operatorname{grad} f(\xi)^{\top}(x-x_0)
    \end{equation}
  \end{YetAnotherTheorem}
\end{document}

edit

Experimenting with the options --

 {enhanced,arc=1mm,outer arc=1mm,
  boxrule=0mm,toprule=0mm,bottomrule=0mm,left=1mm,right=1mm,leftrule=2pt,
  titlerule=0mm,toptitle=0mm,bottomtitle=0mm,top=0mm,
  colframe=blue!50!black,colback=blue!5!white,coltitle=blue!50!black,
  colbacktitle=yellow!50!white,colback=green!5!white,
  title=#1,
  fonttitle=\bfseries\sffamily\normalsize,fontupper=\normalsize\itshape,
  }

--will give different results such as below

edit for title box--by changing the options again

  {
    enhanced,
arc=1mm,
outer arc=1mm,
boxrule=0mm,
toprule=0mm,
bottomrule=0mm,
left=1mm,
right=1mm,
leftrule=2pt,
titlerule=0mm,
toptitle=0mm,
bottomtitle=0mm,
top=0mm,
colframe=blue!50!black,
colback=blue!5!white,
  title=#1,
  fonttitle=\bfseries\sffamily\normalsize,fontupper=\normalsize\itshape,
  attach boxed title to top left={
                                xshift=-1mm,
                                yshift=-5mm,
                                yshifttext=-1mm
                                },  
top=1.5em,
boxed title style={
                      sharp corners,
                      size=small,
                      colback=blue!75!black,
                      colframe=blue!75!black,colbacktitle=white
                    } 
  }