100.sagews

m,L=var('m,L')
x1=(m - L/sqrt(m^2+1))/2
x2=L/sqrt(m^2+1)+x1
y2=x2^2
y1=x1^2
h=(y2+y1)/2
view(expand(h))
hd=expand(diff(h,m))
view(factor(hd))
view(solve(hd==0,m))
plot3d(h, (m,-4,4), (L,0,4))

\frac{{L}^{2} }{{4 {m}^{2} } + 4} + \frac{{m}^{2} }{4}

\frac{{{-m \left( L - {m}^{2} - 1 \right)} \left( L + {m}^{2} + 1 \right)}}{{2 {\left( {m}^{2} + 1 \right)}^{2} }}

$$\begin{array}{l}[m = -\left( \sqrt{ L - 1 } \right),\\ m = \sqrt{ L - 1 },\\ m = -\left( \sqrt{ -L - 1 } \right),\\ m = \sqrt{ -L - 1 },\\ m = 0]\end{array}$$

m=var('m')
L=2
x1=(m - L/sqrt(m^2+1))/2
x2=L/sqrt(m^2+1)+x1
y2=x2^2
y1=x1^2
h=(y2+y1)/2
plot(h,(m,-2,2))

L=var('L')
m=sqrt(L-1)
x1=(m - L/sqrt(m^2+1))/2
x2=L/sqrt(m^2+1)+x1
y2=x2^2
y1=x1^2
h=(y2+y1)/2
expand(h)

L/2 - 1/4