Ttdfjt

What packages can I use and what code to draw these functions?

Some PSTricks solutions only for fun purposes!

documentclass[pstricks,border=12pt,12pt]standalone
usepackagepst-plot,pst-eucl
deff(x-1)^2/5+1
defL#1#2#30,0)$#2mathstrut$uput[180](0,0
begindocument
beginpspicture[algebraic,saveNodeCoors,NodeCoorPrefix=N](-2,-1)(7,5)
 psaxes[labels=none,ticks=none]->(0,0)(-1,-1)(6.5,4.5)[$x$,0][$y$,90]
 psplot[linecolor=red]-15f
 pstGeonode[PosAngle=90](*1 f)P(*3.5 f)Q
 psdot(Q|P)
 pcline[nodesep=-2](P)(Q)
 LPxf(x)
 LQx+varepsilonf(x+varepsilon)
 pcline[linecolor=blue](P)(Q|P)nbput$varepsilon$
 pcline[linecolor=blue](Q)(!NQx NPy)naput$f(x+varepsilon)-f(x)$
 uput[-45]([nodesep=-1]pQ)secant
 uput[0](*5 f)textcolorred$y=f(x)$
endpspicture
enddocument

documentclass[pstricks,border=12pt,12pt]standalone
usepackagepstricks-add,pst-eucl


deff(#1)((#1+3)/3+sin(#1+3))
deffp(#1)Derive(1,f(#1))
pssetunit=2

begindocument
multidor=2.0+-.119%
beginpspicture[algebraic](-1.6,-.6)(4.4,3.4)
 psaxes[ticks=none,labels=none]->(0,0)(-1.6,-.6)(4.1,3.1)[$x$,0][$y$,90]
 psplot[linecolor=red,linewidth=2pt]-13.9f(x)
 %
 psplotTangent[linecolor=blue]1.61f(x)
 psplotTangent[linecolor=cyan,Derive=-1/fp(x)]1.6.5f(x)
 %
 pstGeonode[PosAngle=135,90]
 (*1.6 f(x))A
 (*1.6 rspace add f(x))B
 pstGeonode[PosAngle=-120,-60,PointName=x_1,x_2,PointNameSep=8pt]
 (A
enddocument

I recommend TikZ for that. (I used to love pstricks, and the pstricks solution is really neat and I upvoted it, but having seen what TikZ can do I can no longer recommend pstricks, sorry.)

documentclass[tikz,border=3.14mm]standalone
usetikzlibraryintersections
begindocument
begintikzpicture[declare function=f(x)=0.3*(x-3.5)^3-x+7;a=1;b=6;c=4.94;]
 draw[-stealth] (-0.5,0) -- (6.5,0);
 draw[-stealth] (0,-0.5) -- (0,6.5);
 draw[blue] plot[smooth,domain=0.5:6.1] (x,f(x));
 foreach X in a,b
 - (0,f(X)) node[left] $f(X)$;
 draw (a,f(a)) -- (b,f(b));
 draw[dashed] (c,0) -- (c,f(c));
 draw[dashed,name path=hori] (a,f(a)) -- (b,f(a));
 pgfmathsetmacroslopeangleatan2(f(b)-f(a),b-a)
 draw[red,name path=sloped] (c,f(c)) +(slopeangle:2) -- ++ (slopeangle+180:4);
 draw (a,f(a)) + (1,0) arc(0:slopeangle:1) node[midway,right]$beta$;
 draw[name intersections=of=hori and sloped,by=i] (i) +(1,0)
 arc(0:slopeangle:1) node[midway,right]$beta$;
endtikzpicture
enddocument

Adding a MetaPost solution, for completeness. This is how we did it in a text we write for students. Since I prefer not to put too many labels in the figures, I rather explain in text that the "dashed lines are parallell, and hence ..."

As it is written, one can run context on the file, but one can easily adopt it to be plain MetaPost.

startMPpage
%Set unit
u=1cm;

%Introduce paths
path p,xax,yax;

% Draw axes
xax = ((-0.5,0)--(7.5,0));
yax = ((0,-0.5)--(0,4));
drawarrow xax scaled u;
drawarrow yax scaled u;

%Define your path p
z0 = (1.5u,u);
z1 = (3u,3u);
z2 = (5u,3u);
z3 = (6.5u,2u);

p = z0dir 80..z1..dir 0z2..dir -10z3;

%Find the right "time" and tangent point (calculated by MetaPost)
t = directiontime (z3-z0) of p;
z4 = point t of p;

%Draw path, secant and tangent
draw p;
draw z0--z3 dashed evenly;
draw (z0--z3) shifted (z4-0.5[z0,z3]) dashed evenly;

label.bot(textext("$(a,f(a))$"), z0);
label.lrt(textext("$(b,f(b))$"), z3);
label.ulft(textext("$(xi,f(xi))$"), z4);
stopMPpage

The result looks like this:

Some more fun with pstricks, which has a psPlotTangent command:

documentclass[svgnames, x11names, border = 5pt]standalone%
usepackage[utf8]inputenc
usepackageamsmath 
 usepackageauto-pst-pdf%
usepackagepstricks-add%,
defFx^3-6*x^2 + 9*x + 1

begindocument

pssetunit=2cm, arrowinset=0.12, algebraic, plotstyle=curve, plotpoints=200, dimen=inner
everypsboxfootnotesize
beginpspicture*(-1,-1)(6,5.5)
psaxes[linecolor = LightSteelBlue, ticks=none, labels=none]->(0,0)(-2,-1.2)(5,5.5)[$x$,-135][$y$,-135]
 psplot[linecolor = IndianRed, linewidth =1.2pt]0.054F
pssetlinestyle=dashed, linewidth=0.3pt
psCoordinates(*0.5 F)uput[d](0.5,0)$a$uput[l](0,4.125)$f(a)$)
psCoordinates(*3.5 F)uput[d](3.5,0)$b$uput[l](0,1.875)$f(b)$)
psline[linecolor=Gold, linewidth=0.6pt] (0.5, 4.125)(3.5,1.875)
psline(1.134,0)(1.134, 4.949)(3.134, 4.949)uput[d](1.134,0)$c$
psline(2.866, 0)(2.866, 1.051)(4.866,1.051)uput[d](2.866,0)$c_1$
pssetlinestyle=solid, labelsep=24pt
foreach x in 1.134, 2.866psplotTangent[algebraic, linewidth=0.6pt, Derive=3*x^2-12*x + 9, linecolor=Gold, showpoints]x1.5F
psarc(3.5, 1.875)0.4143180uput[161](3.5, 1.875)$beta$
psarcn(1.134, 4.949)0.40-37uput[-18](1.134, 4.949)$beta$
psarcn(2.866, 1.051)0.40-37uput[-18](2.866, 1.051)$beta$
rput(5,1.5)$boxedtanbeta = dfracf(b)-f(a)b-a = f'(c)$
endpspicture*

enddocument