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Parameter values for which a function is monotonic or has no critical point

This type of question, asked 5 times: 2021 Paper 5, 2021 Paper 22, 14 Jun 2022, 26 Jul 2022, 9 Apr 2024. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    74
    The range of a∈Ra\in\mathbb{R} for which the function f(x)=(4a−3)(x+log⁡e5)+2(a−7)cot⁡(x2)sin⁡2(x2)f(x) = (4a- 3)\left( x+\log_{e}5 \right)+ 2(a- 7)\cot\left( \frac{x}{2} \right)\sin^{2}\left( \frac{x}{2} \right), x≠2nπ,n∈Nx\neq 2n\pi,n\in\mathbb{N}, has critical points, is:

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  2. Question 2

    2021 Paper 22

    84
    If ' RR ' is the least value of ' aa ' such that the function f(x)=x2+ax+1f(x) =x^{2}+ax+ 1 is increasing on [1, 2] and 'S' is the greatest value of ' aa ' such that the function f(x)=x2+ax+1f(x) =x^{2}+ax+ 1 is decreasing on [1,2]\lbrack 1,2\rbrack, then the value of ∣R−S∣|R - S| is

  3. Question 3

    14 Jun 2022

    170
    Let λ∗\lambda^{*} be the largest value of λ\lambda for which the function fλ(x)=4λx3−36λx2+36x+48f_{\lambda}(x) = 4\lambda x^{3}- 36\lambda x^{2}+ 36x + 48 is increasing for all x∈Rx \in R. Then fλ∗(1)+fλ∗(−1)f_{\lambda}*(1) +f_{\lambda}*( - 1) is equal to:

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  4. Question 4

    26 Jul 2022

    156
    If the maximum value of aa, for which the function fa(x)=tan⁡−12x−3ax+7f_{a}(x) =\tan^{- 1}2x - 3ax + 7 is non-decreasing in (−π6,π6)\left( -\frac{\pi}{6},\frac{\pi}{6} \right), is aˉ\bar{a}, then faˉ(π8)f_{\bar{a}}\left( \frac{\pi}{8} \right) is equal to

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  5. Question 5

    9 Apr 2024

    172
    Let the set of all values of pp, for which f(x)=(p2−6p+8)(sin⁡22x−cos⁡22x)+2(2−p)x+7f(x) =\left( p^{2}- 6p + 8 \right)\left( \sin^{2}2x -\cos^{2}2x \right)+ 2(2 - p)x + 7 does not have any critical point, be the interval (a,b)(a,b). Then 16ab16ab is equal to