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Cubic with a parameter: condition on its local max and min points

This type of question, asked 3 times: 8 Apr 2024, 2 Apr 2025, 4 Apr 2025. All 3 questions below, oldest first.

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  1. Question 1

    8 Apr 2024

    164
    If the function f(x)=2x3−9ax2+12a2x+1,a>0f(x) = 2x^{3}- 9ax^{2}+ 12a^{2}x+ 1,a> 0 has a local maximum at x=αx =\alpha and a local minimum x=α2x=\alpha^{2}, then α\alpha and α2\alpha^{2} are the roots of the equation :

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  2. Question 2

    2 Apr 2025

    59
    If the function f(x)=2x3−9ax2+12a2x+1f(x) = 2x^{3}- 9ax^{2}+ 12a^{2}x+ 1, where a >0> 0, attains its local maximum and local minimum values at pp and qq, respectively, such that p2=qp^{2}= q, then f(3)f(3) is equal to:

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  3. Question 3

    4 Apr 2025

    126
    Let a>0a > 0. If the function f(x)=6x3−45ax2+108a2x+1f(x) = 6x^{3}- 45ax^{2}+ 108a^{2}x + 1 attains its local maximum and minimum values at the points x1x_{1} and x2x_{2} respectively such that x1x2=x_{1}x_{2}= 54 , then a+x1+x2a +x_{1}+x_{2} is equal to :-

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