ϕSol=ϕ/.ParametricNDSolve[ {
ϕ''[x]+ λ ϕ[x]==0,
ϕ[0.5]==1,
ϕ'[0.5]==0
},
ϕ,
{x,0,1},
{λ}];

λVal=123.1;
Plot[ϕSol[λVal][x],{x,0,1}]

Plot[ϕSol[λ]'[1],{λ,0, 123}]
FindRoot[ϕSol[λ]'[1]==0, {λ,123}]
Plot[ϕSol[λ]'[1],{λ,0, 123}]