How to find the intervals in which a function is positive?

Question

I have to find the positive range for the derivatives of a lot of functions. I have a very inefficient solution.

At the moment I am using a For loop to find the times where the function is positive and use Split as used in #23608 to find the intervals.

Full code:

f[t_] := Sqrt[0.9604 + 0.0099 (Cos[1. t] + Cos[3.14159 t])^2]
ptimes = {};
inc = 0.01;
For[pt = 0, pt <= 10, pt = pt + inc, 
    If[(D[f[z],z]/.{z -> pt}) >= 0, AppendTo[ptimes, pt] && Continue[], Continue[]]
   ]
int = {Min[#], Max[#]} & /@ Split[ptimes, #2 - #1 <= inc &]
({{0, 0}, {0.76, 1.09}, {1.47, 1.9}, {2.28, 3.01}, {3.8, 4.}, {4.01, 
  4.08}, {4.41, 4.89}, {5.31, 6.02}, {6.83, 7.07}, {7.34, 
  7.89}, {8.35, 9.03}, {9.87, 10.}})

Although this works, but I am sure there are efficient ways to do this. Appreciate your thoughts.

Note: Using Reduce or Solve barely works for me when some of the arguments inside are irrational.

Answer 1

After replacing 3.14159 with $\pi$ it works:

f[t_] = Sqrt[0.9604 + 0.0099 (Cos[t] + Cos[π*t])^2];
Reduce[f'[t] > 0 && -10 <= t <= 10, t]
(*    -9.86111  < t < -9.03828 || -8.34402  < t < -7.89698 ||
      -7.33471  < t < -7.07256 || -6.82692  < t < -6.02582 ||
      -5.30983  < t < -4.89873 || -4.40083  < t < -4.08283 ||
      -3.79273  < t < -3.013   || -2.27564  < t < -1.9027  ||
      -1.46694  < t < -1.09113 || -0.758547 < t <  0       ||
       0.758547 < t <  1.09113 ||  1.46694  < t <  1.9027  ||
       2.27564  < t <  3.013   ||  3.79273  < t <  4.08283 ||
       4.40083  < t <  4.89873 ||  5.30983  < t <  6.02582 ||
       6.82692  < t <  7.07256 ||  7.33471  < t <  7.89698 ||
       8.34402  < t <  9.03828 ||  9.86111  < t <= 10.          *)

Answer 2

Try this:

f[t_] := Sqrt[0.9604 + 0.0099 (Cos[1. t] + Cos[\[Pi] t])^2] // 
  Rationalize
g[t_] := D[f[t], t]
Reduce[g[t] >= 0 && 0 < t < [Pi]] // N
(*  0.758547 <= t <= 1.09113 || 1.46694 <= t <= 1.9027 || 
 2.27564 <= t <= 3.013   *)

Have fun!