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Absolute max and min of a polynomial on a closed interval

This type of question, asked 3 times: 30 Jan 2024, 22 Jan 2026 S1, 6 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    30 Jan 2024

    155
    Let f(x)=(x+3)2(x−2)3,x∈[−4,4]f(x) = (x+ 3)^{2}(x- 2)^{3},x\in \lbrack - 4,4\rbrack. If MM and mm are the maximum and minimum values of ff, respectively in [−4,4]\lbrack - 4,4\rbrack, then the value of M−mM-m is:

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  2. Question 2

    22 Jan 2026 S1

    55
    Let f(x)=x2025−x2000,x∈[0,1]f(x) =x^{2025}-x^{2000},x \in \lbrack 0,1\rbrack and the minimum value of the function f(x)f(x) in the interval [0,1]\lbrack 0,1\rbrack be (80)80(n)−81(80)^{80}(n)^{- 81}. Then n is equal to

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  3. Question 3

    6 Apr 2026 S2

    67
    Let A=∣13−121α01−1∣A =\left| \begin{matrix} 1 & 3 & - 1 \\ 2 & 1 & \alpha \\ 0 & 1 & - 1 \end{matrix} \right| be a singular matrix. Let f(x)=∫0x(t2+2t+3)dt,x∈[1,α]f(x) =\int_{0}^{x} \left( t^{2}+ 2t+ 3 \right)dt,x\in \lbrack 1,\alpha\rbrack. If MM and mm are respectively the maximum and the minimum values of ff in [1,α]\lbrack 1,\alpha\rbrack then 3(M−m)3(M-m) is equal to :

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