read(`fenchel.m`);

# Example 1.

f1 := convert(abs(x),cf);

subdiff(f1);


g1 := conj(f1,y);

# Example TWO
f2 := convert(exp(x),cf);

g2 := conj(f2,y);

conj(g2,x);

# Example THREE
f3 := cf([[-infinity,1/2*(x^2-4*x+3),1],
          [1,0,1],[1,-ln(x),infinity]],x);

g3 := conj(f3,y);


# Example FOUR
convert(b*x+c,cf,x);

conj(",y);

assume(a>0);
convert(a*x^2+b*x+c,cf,x);

conj(",y);

# Example FIVE
f5 := convert(ln((1/2)*(2^x+(2/3)^x)),cf);

g5 := conj(",y): # conjugate is big
cfeval(g5,y=1/2*ln(4/3)); 

temp := op([1,3,2],g5):
ybar := fsolve(temp=y,y);

# Example SIX
f6 := cf([[-infinity,exp(-4*x-2),-1/2], 
          [-1/2,1,-1/2],
          [-1/2,-x/(1+x),infinity]],x);

factor(conj(f6,y));

# Example SEVEN
f7 := convert(piecewise(x>=0,
               a*x+x^2,0),cf,x);

g7 := factor(conj(f7,y));

# Example EIGHT
f8 := convert(piecewise(
       -3<=x and x<=1,
       abs(x)-2*sqrt(1-x),
       infinity),cf);


cfplot(f8,x=-4..2,scaling=constrained,
       axes=framed);
sdf8 := subdiff(f8);

sdplot(sdf8,x=-3..1,view=[-3..1,-3..5],
       axes=none);
g8 := conj(f8,y);

cfplot(g8,y=-3..3,scaling=constrained);

simplify(convpw(g8));
# Example NINE
f1;
f9 := convert(1/2*x^2,cf);

simplify(conj(f1,y) + conj(f9,y));

conj(",x);


#Example TEN
f10 := convert(1/3*abs(x)^3,cf);


g10 := conj(f10,y);