x=var('x')
y=var('y')
z=var('z')
f(x,y,z)=x^3+y^3+z^3-5*(x^2*y+y^2*z+z^2*x)+4*(x*y^2+y*z^2+z*x^2)
print 'global derivative',diff(f(x,y,z),x)+diff(f(x,y,z),y)+diff(f(x,y,z),z)
g(x,y,z)=diff(f(x,y,z),x)
print 'f_x(20,10,1)=',g(20,10,1)
g(x,y,z)=diff(f(x,y,z),y)
print 'f_y(20,10,1)=',g(20,10,1)
g(x,y,z)=diff(f(x,y,z),z)
print 'function is increasing in range (0,1):'
print 'f_z(20,10,1)=', g(20,10,1)
print 'f(20,10,1=',f(20,10,1)
print 'f(20,10,0)=',f(20,10,0)
print 'consider the case f(20,10,0) to end the emv proof:'
print 'f(x,1,0)=',f(x,1,0)
show(plot(f(x,1,0),0,4))
print 'failed to prove: f(2,1,0) is negative'
global derivative 2*x^2 - 2*x*y - 2*x*z + 2*y^2 - 2*y*z + 2*z^2
f_x(20,10,1)= -245
f_y(20,10,1)= -196
function is increasing in range (0,1):
f_z(20,10,1)= 983
f(20,10,1= -1959
f(20,10,0)= -3000
consider the case f(20,10,0) to end the emv proof:
f(x,1,0)= x^3 - 5*x^2 + 4*x + 1
failed to prove: f(2,1,0) is negative
