How can I make a general plot like this in Mathematica?

Question

I want to make two general graphics something like the following.

I want to illustrate the approach to the function by means of a polynomial. I have tried the following.

Plot[{Sin[x] + 2,  Sin[2 x] + 2}, {x, 0, 8}, AxesLabel -> {x, y}, 
Ticks -> {{0.8, 2.4, 4, 5.5, 7.1}, {0, 1, 2, 3}}]

I also want to put $ x_{0}, x_{1}, x_{2},...,x_{n} $ in the points where the graphs differ the most. I appreciate any help.

Answer 1

f[x_] := 1 + x + Sin[10 Sqrt[1 + x]];
g[x_] := 1 + x + Cos[10 Sqrt[2 + x]];
xx = x /. NSolve[{f'[x] == g'[x], 0 <= x <= 8}, x];
ticks = MapIndexed[{#, Style[ Subscript["x", ToString@ #2[[1]]],  16]} &, xx];
Plot[{f[x], g[x]}, {x, 0, 3 Pi}, AxesLabel -> {x, y}, 
 MeshFunctions -> {f'[#] - g'[#] &}, Mesh -> {{0}}, 
 MeshStyle -> Directive[Red, PointSize[Large]], 
 GridLines -> {xx, None}, Ticks -> {ticks, Automatic}, 
 AxesOrigin -> {0, 0}]

Answer 2

Something like this?

    Plot[{Sin[x] + 2, Sin[2 x] + 2}, {x, 0, 8}, AxesLabel -> {x, y}, 
 Ticks -> {{{1, "x[0]"}, {2, "x[1]"}, {3, ""}, {4, "x[3]"}, {5, 
     ""}, {6, ""}, {7, "x[n]"}}, None}, 
 PlotLabels -> Placed[{"y=f(x)", "y=p(x)"}, {Scaled[5], Above}], 
 AxesStyle -> Arrowheads[{0.0, 0.05}]]

Or

Plot[{Sin[x] + 2, Sin[2 x] + 2}, {x, 0, 8}, AxesLabel -> {x, y}, 
 Ticks -> {{{1, "x[0]"}, {2, "x[1]"}, {3, ""}, {4, "x[2]"}, {5, 
     ""}, {6, ""}, {7, "x[n]"}}, None}, 
 AxesStyle -> Arrowheads[{0.0, 0.05}], 
 Epilog -> {Text[Style["y=f(x)", 22], Scaled[{0.5, 0.9}]], 
   Text[Style["y=p(x)", 22], Scaled[{0.25, 0.97}]]}]