Mfcttrf

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#3#1)$#3$
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

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