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Minimum value using AM-GM or calculus on a sum of two terms

This type of question, asked 3 times: 2021 Paper 1, 2021 Paper 4, 28 Jul 2022. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    83
    The minimum value of α\alpha for which the equation 4sin⁡x+11−sin⁡x=α\frac{4}{\sin x}+\frac{1}{1 - \sin x}=\alpha has at least one solution in (0,π2)\left( 0,\frac{\pi}{2} \right) is

  2. Question 2

    2021 Paper 4

    73
    The minimum value of f(x)=aax+a1−axf(x) =a^{a^{x}}+a^{1 -a^{x}}, where a,x∈Ra,x\in R and a>0a> 0, is equal to:

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  3. Question 3

    28 Jul 2022

    77
    If the minimum value of f(x)=5x22+αx5,x>0f(x) =\frac{5x^{2}}{2}+\frac{\alpha}{x^{5}},x > 0, is 14, then the value of α\alpha is equal to:

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