5. The energy transfer is increasing at the fastest rate when its change (derivative) is maximized. That happens when the second derivative is zero while the first derivative is positive.

Input := 

sol = Solve[f''[x] == 0, x]

Output =

{{x -> 0.00109607}, {x -> 0.0315948}, 
 
  {x -> 0.0707862}, {x -> 0.122078}, 
 
  {x -> 0.176662}}

Input := 

xs = {x /. sol[[1, 1]], x /. sol[[2, 1]],
      x /. sol[[3, 1]], x /. sol[[4, 1]],
      x /. sol[[5, 1]]}

Output =

{0.00109607, 0.0315948, 0.0707862, 
 
  0.122078, 0.176662}

Input := 

{f'[xs[[1]]], f'[xs[[2]]], f'[xs[[3]]],
 f'[xs[[4]]], f'[xs[[5]]]}

Output =

{1.75891, 67.7416, 2.89308, 165.359, 
 
  -408.922}

Of the candidates, we see that when x = 0.122078, that f(x) is increasing the fastest at a rate of 165.359 hp/second.