1948.sagews

sage: f(x)=x^4-(4*x^2)

f.diff(x); diff(f,x,2)

x |--> 4*x^3 - 8*x x |--> 12*x^2 - 8

df(x)=4*x^3-8*x
solve(df==0,x)

[x == -sqrt(2), x == sqrt(2), x == 0]

plot(f,-5,5)

solve((12*x^2 - 8)==0,x);

[x == -1/3*sqrt(2)*sqrt(3), x == 1/3*sqrt(2)*sqrt(3)]

sage: g=4/((x^2)-1)

g.diff(x); diff(g,x,2)

-8*x/(x^2 - 1)^2 32*x^2/(x^2 - 1)^3 - 8/(x^2 - 1)^2

dg(x)=-8*x/(x^2 - 1)^2
solve(df==0,x)

[x == -sqrt(2), x == sqrt(2), x == 0]

plot(g,-1.5,1.5)

solve((32*x^2/(x^2 - 1)^3 - 8/(x^2 - 1)^2)==0,x);

[x == -1/3*I*sqrt(3), x == 1/3*I*sqrt(3)]