Plot surface of constraints: Possibly via Apply or Map Reduce over a list of equations

Question

I have a function f(A,B,C) where for specific A and B values I can use Reduce to determine the constraint on C for my problem using a constraint on f. I want to plot the resulting surface.

To illustrate, consider

f = A^5 + B^3 + C^2

If A and B vary between 1 and 3 then I get the list of constraints (with f<20)

constraints = {{1, 1, C < 18}, {2, 1, C < -13}, {3, 1, C < -224}, {1, 2, C < 11}, {2, 2, C < -20}, {3, 2, C < -231}, {1, 3, C < -8}, {2, 3, C < -39}, {3, 3, C < -250}}

I then want to plot the surface given by

surf = {{1, 1, 18}, {2, 1, -13}, {3, 1, -224}, {1, 2, 
  11}, {2, 2, -20}, {3, 2, -231}, {1, 3,  -8}, {2, 3, 
   -39}, {3, 3, -250}}

ListPlot3D[surf,Mesh->All]

I can form the list of constraints using For loops

constraints = {};
For[B = 1, B <= 3, B++,
 For[A = 1, A <= 3, A++,
  f = (A)^5 + B^3 + p;
  sol  = Reduce[f < 20, p];
  constraints = Append[constraints, {A, B, sol}]
  ]
 ]
constraints

However I am not sure how to get from the list of constraints to the max permitted value for C and therefore get to the surf expression.

I also expect that For loops are not an ideal approach, and that I should be able to form lists of the A and B values and use another approach (Map, or Thread, or Apply maybe) with Reduce. I find these methods confusing though, and don't really understand anything but the most basic examples (so possibly similar questions have not helped me figure this out).

Answer 1

f = a^5 + b^3 + c^2
RegionPlot3D[f <= 20, {a, 1, 3}, {b, 1, 3}, {c, -5, 5}, 
 AxesLabel -> {"a", "b", "c"}]