Consider the following plot:
Plot[(22500 + T^2/3) ϕ^2 + 320 ϕ^3 + ϕ^4 /. T -> 96.43650760992945`, {ϕ, -200, 20}]
I would like to find the value of $\Phi<0$ and $T>0$ to obtain the first minimum of the above function.
These two values can be obtained with
FindRoot[
{(22500 + T^2/3) ϕ^2 + 320 ϕ^3 + ϕ^4 == 0 ,
D[(22500 + T^2/3) ϕ^2 + 320 ϕ^3 + ϕ^4, {ϕ, 1}] == 0},
{{ϕ, -200}, {T, 1}}]
which yields
FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances.
{ϕ -> -160., T -> 96.4365}
However, I would like to obtain it with NSolve but it does not work:
NSolve[
{(22500 + T^2/3) ϕ^2 + 320 ϕ^3 + ϕ^4 == 0,
D[(22500 + T^2/3) ϕ^2 + 320 ϕ^3 + ϕ^4, {ϕ, 1}] == 0},
{ϕ, T}]
which gives
NSolve::infsolns: Infinite solution set has dimension at least 1. Returning intersection of solutions with -((92291 T)/87992)-(121001 [Phi])/175984 == 1.
{{ϕ -> 0., T -> -0.953419}}
as a solution, which is obviously wrong.