g: ℝ → ℝ, g↑
(fog)′(x) = f ′g(x) . g ′(x) = (≤ 0). (≥ 0)
⇒ (fog)′ (x) ≤ 0
⇒ fog(x) is ↓
(gof)′ (x) = g ′(f(x) . f ′(x) = (≥ 0). (≤ 0)
⇒ (gof)′ (x) ≤ 0
⇒ gof(x) is ↓
(fof)′ (x) = f ′(f(x)) . f ′(x) = (≤ 0) (≤ 0)
⇒ (fof)′ (x) ≥ 0
⇒ fof(x) is ↑ and (gog)′ (x) = g ′(g(x)). g ′(x) = (≥ 0). (≥ 0)
⇒ gog(x) is ↑
Thus fof and gog are increasing functions.
⇒ f′(x)
⇒ f(x) increase for ad ≥ bc
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