How to test whether a function is positive over the entire range of an interval?

To find the intervals for which f[x] is positive

f[x_] = -x^3 + x^2 + 7*x;

g[x_] = Piecewise[{{f[x], f[x] > 0}}, I];

Plot[{f[x], g[x]}, {x, -3, 4},
 PlotStyle -> {Directive[Red, Dashed], Blue}]

FunctionDomain[g[x], x]

(*  x < (1/2)*(1 - Sqrt[29]) || 
   0 < x < (1/2)*(1 + Sqrt[29])  *)

% // N

(*  x < -2.19258 || 0. < x < 3.19258  *)

EDIT:

Or, more succinctly

FunctionDomain[Piecewise[{{1, f[x] > 0}}, I], x]

(*  x < (1/2)*(1 - Sqrt[29]) || 
   0 < x < (1/2)*(1 + Sqrt[29])  *)

I like the answers using Reduce and FunctionDomain. Here's a numerical possibility that uses Minimize to find the global minimum on the domain and tests to see if it's positive.

f[x_] = -x^3 + x^2 + 7*x;
0 <= First@Minimize[{f[x], 0 <= x <= 4}, x]
(* False *)

Alternatively, if needed, you can use NMinimize:

NMinimize[{f[x], 0 <= x <= 4}, x]
(* {-20., {x -> 4.}} *)
0 <= First@NMinimize[{f[x], 0 <= x <= 4}, x]
(* False *)