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Polynomial found from its extrema conditions

This type of question, asked 6 times: 2021 Paper 3, 2021 Paper 10, 2021 Paper 23, 25 Jul 2022, 7 Apr 2025, 2 Apr 2026 S2. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    85
    Let f(x)f(x) be a polynomial of degree 66 in xx in which the coefficient of x6x^6 is unity and which has extrema at x=−1x=-1 and x=1x=1. If lim⁡x→0f(x)x3=1\lim_{x \to 0} \dfrac{f(x)}{x^3} = 1, then 5 f(2)5\,f(2) is equal to

  2. Question 2

    2021 Paper 10

    85
    Let P(x)P(x) be a real polynomial of degree 3 which vanishes at x=−3x= - 3. Let P(x)P(x) have local minima at x=1x = 1, local maxima at x=−1x = - 1 and ∫−11P(x)dx=18\int_{- 1}^{1} P(x)dx = 18, then the sum of all the coefficients of the polynomial P(x)P(x) is equal to

  3. Question 3

    2021 Paper 23

    90
    Let f(x)f(x) be a cubic polynomial with f(1)=−10f(1) = - 10, f(−1)=6f( - 1) = 6, and has a local minima at x=1x = 1, and f′(x)f^{'}(x) has a local minima at x=−1x = - 1. Then f(3)f(3) is equal to

  4. Question 4

    25 Jul 2022

    67
    The curve y(x)=ax3+bx2+cx+5y(x) = ax^{3}+ bx^{2}+ cx + 5 touches the xx-axis at the point P(−2,0)P( - 2,0) and cuts the yy-axis at the point QQ, where y′y^{'} is equal to 3 . Then the local maximum value of y(x)y(x) is:

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  5. Question 5

    7 Apr 2025

    133
    Let f:R→Rf:R \rightarrow R be a polynomial function of degree four having extreme values at x=4x = 4 and x=5x = 5.If lim⁡x→0f(x)x2=5\lim_{x \rightarrow 0} \frac{f(x)}{x^{2}}= 5, then f(2)f(2) is equal to :

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  6. Question 6

    2 Apr 2026 S2

    68
    Let f(x)f(x) be a polynomial of degree 5 and have extrema at x=1x= 1 and x=−1x= - 1. If lim⁡x→0(f(x)x3)=−5\lim_{x\rightarrow 0} \left( \frac{f(x)}{x^{3}} \right)= - 5, then f(2)−f(−2)f(2) - f( - 2) is equal to :

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