Cfrgtkky

The extra y ticks at y = 6000 and y = 5400 and the extra x ticks at x = 24 and x = 20.4 are not drawn.

    documentclass[border=10pt]{standalone}

    usepackage{pgfplots}

    usetikzlibrary{intersections}

    usepgfplotslibrary{fillbetween}

    pgfplotsset{major grid style={thick,darkgray}}

    pgfplotsset{minor grid style={lightgray}}

    begin{document}

    begin{tikzpicture}[

     extended line/.style={shorten >=-#1,shorten <=-#1},

    /pgf/declare function={f(x)=  1000*(x +5)*exp(-0.2*x);}]

    begin{axis}[

         restrict x to domain=0:22, xmax=22, xmin=0,

         restrict y to domain=0:6000, ymax=6000, ymin=0,

         x=0.6cm,

         y=0.0012cm,

         axis x line=bottom,

         axis y line=left,

         major tick style=black,

         axis line style = ultra thick,

         tick align=outside,

         tickwidth=0.14cm,

         tick style=thick,

         minor x tick num=4,

         minor y tick num=4,

         grid=both,

         xtick={0,2,...,20},

         ytick={0,1000,...,5000},

         y tick label style={/pgf/number format/1000 sep={}},

         extra x ticks={20.4,20.8,21.2,21.6,22},

         extra x tick label={null},

         extra y ticks={5200,5400,5600,5800,6000},

         extra y tick label={null},

         extra x tick style={grid=minor,tick style={draw=none}},

         extra y tick style={grid=minor,tick style={draw=none}},

         xlabel=$x$,ylabel=$f(x)$,

         every axis x label/.style={

         at={(ticklabel* cs:1.01)},anchor=west},

         every axis y label/.style={

         at={(ticklabel* cs:1.01)}, anchor=south},

         samples=3000,

         axis on top=false,

         >=stealth,

         ]

        addplot [ultra thick,domain=0:20] {f(x)};

        end{axis}                  

        end{tikzpicture}

        end{document}

if i understood correctly what you like to achieve, than you not need extra ticks. you only need to define tick labels accordingly:

documentclass[border=10pt]{standalone}

usepackage{pgfplots}

usetikzlibrary{intersections}

usepgfplotslibrary{fillbetween}



pgfplotsset{major grid style={thick,darkgray}}

pgfplotsset{minor grid style={lightgray}}

begin{document}

begin{tikzpicture}[

 extended line/.style={shorten >=-#1,shorten <=-#1},

/pgf/declare function={f(x)=  1000*(x +5)*exp(-0.2*x);}]

begin{axis}[

     xmax=22, xmin=0,

     ymax=6000, ymin=0,

     x=0.6cm,

     y=0.0012cm,

     axis x line=bottom,

     axis y line=left,

     major tick style=black,

     axis line style = ultra thick,

     tick align=outside,

     tickwidth=0.14cm,

     tick style=thick,

     minor x tick num=4,

     xtick={0,2,...,22},

     xticklabels = {0,2,...,20,},         % <----

     ytick={0,1000,...,6000},

     yticklabels={0,1000,2000,...,5000,},  % <----

     xlabel=$x$,ylabel=$f(x)$,

     minor y tick num=4,

     grid,

     samples=40,

     >=stealth,

     ]

    addplot [ultra thick,domain=0:20] {f(x)};

    end{axis}

    end{tikzpicture}

    end{document}

xtick={0,2,...,22},
ytick={0,1000,...,6000} should allow what you need

unrelated to your question add also pgfplotsset{compt=1.15} int the preamble

documentclass[border=10pt]{standalone}

 usepackage{pgfplots}

 usetikzlibrary{intersections}

  usepgfplotslibrary{fillbetween}

  pgfplotsset{compt=1.15} % <-------

  pgfplotsset{major grid style={thick,darkgray}}

  pgfplotsset{minor grid style={lightgray}}

begin{document}

begin{tikzpicture}[

 extended line/.style={shorten >=-#1,shorten <=-#1},

/pgf/declare function={f(x)=  1000*(x +5)*exp(-0.2*x);}]

begin{axis}[

     restrict x to domain=0:22, xmax=22, xmin=0,

     restrict y to domain=0:6000, ymax=6000, ymin=0,

     x=0.6cm,

     y=0.0012cm,

     axis x line=bottom,

     axis y line=left,

     major tick style=black,

     axis line style = ultra thick,

     tick align=outside,

     tickwidth=0.14cm,

     tick style=thick,

     minor x tick num=4,

     minor y tick num=4,

     grid=both,

     xtick={0,2,...,22}, % <--------

     ytick={0,1000,...,6000}, % <-----------

     y tick label style={/pgf/number format/1000 sep={}},

     extra x ticks={20.4,20.8,21.2,21.6,22},

     extra x tick label={null},

     extra y ticks={5200,5400,5600,5800,6000},

     extra y tick label={null},

     extra x tick style={grid=minor,tick style={draw=none}},

     extra y tick style={grid=minor,tick style={draw=none}},

     xlabel=$x$,ylabel=$f(x)$,

     every axis x label/.style={

     at={(ticklabel* cs:1.01)},anchor=west},

     every axis y label/.style={

     at={(ticklabel* cs:1.01)}, anchor=south},

     samples=3000,

     axis on top=false,

     >=stealth,

     ]

    addplot [ultra thick,domain=0:20] {f(x)};

    end{axis}                  

    end{tikzpicture}

    end{document}