have : sqrt (2 * 512) = 32 :=
  (sqrt_eq_iff_mul_self_eq_of_pos (by norm_num1)).mpr (by norm_num1)
case refine'_2
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => x.log + (2 * x).sqrt * (2 * x).log - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ (2 * x).sqrt / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = (x * (2 * x) ^ (2 * x).sqrt / 4 ^ (x / 3)).log
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
⊢ f 512 ≤ 0
case refine'_2
x : ℝ
n_large : 512 ≤ x
f : ℝ → ℝ := fun x => x.log + (2 * x).sqrt * (2 * x).log - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ (2 * x).sqrt / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = (x * (2 * x) ^ (2 * x).sqrt / 4 ^ (x / 3)).log
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f
this : (2 * 512).sqrt = 32
⊢ f 512 ≤ 0