Proceed with the minimization process using Lagrange multiplier

Input := 

e1 = D[ V[V0], x0] == 0;
e2 = D[ V[V0], v0 ] == 0;
e3 = D[ V[V0], lambda] == 0;

Input := 

s = Solve[ {e1,e2,e3}, {x0,v0,lambda}]

Output =

{{lambda -> 0.994988, x0 -> -0.998733, 
 
   v0 -> -0.050313}, 
 
  {lambda -> 0.994988, x0 -> 0.998733, 
 
   v0 -> 0.050313}, 
 
  {lambda -> 1.01519, x0 -> -0.050313, 
 
   v0 -> 0.998733}, 
 
  {lambda -> 1.01519, x0 -> 0.050313, 
 
   v0 -> -0.998733}}

Compare the lambdas in the expression above with the below expression. Recall that A is the matrix representing the difference scheme.

Input := 

Sqrt[ Eigenvalues[ Transpose[A] . A]]

Output =

{1.01519, 0.994988}

Select the maximum and minimum starting vectors. Note that each has two possibilities: one being just the additive inverse of the other. You now have the answer to part (1) of the problem.

Input := 

vmax = {x0,v0} /. s[[3]]

Output =

{-0.050313, 0.998733}

Input := 

Lmax = lambda /. s[[3]]

Output =

1.01519

Input := 

vmin = {x0, v0} /. s[[1]]

Output =

{-0.998733, -0.050313}

Input := 

Lmin = lambda /. s[[1]]

Output =

0.994988