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FUNDAMENTALS OF MATHEMATICS DIFFERENTIAL CALCULUS by Sanjay Mishra

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TEXTUAL EXERCISE–4: (OBJECTIVE)
  1. (a), (b) f: ℝ → ℝ, f

    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.

  2. images

    f′(x)

    images
    images

    f(x) increase for ad ≥ bc

  3. (c) ...

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