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For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by 
S = {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}. 
For a polynomial f, let f ' and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf' + mf"), where f ∈ S, is _______
    Correct answer is '5.00'. Can you explain this answer?
    Verified Answer
    For a polynomial g(x) with real coefficient, let mg denote the number ...
    The minimum possible value of (mf' + mf"), where f ∈ S, is 5. This can be seen as follows:
    Every polynomial in S is of the form (x^2 - 1)^2 * (a0 + a1x + a2x^2 + a3x^3). Since the coefficient of x^3 is non-zero, the polynomial f has at least one real root.

    The first order derivative of f is:

    f' = 2*(x^2 - 1)*(a0 + a1x + a2x^2 + a3x^3) + (a0 + a1x + a2x^2 + a3x^3) * (2x)

    The second order derivative of f is:

    f" = 2*(x^2 - 1)(2x) + (a0 + a1x + a2x^2 + a3x^3) * (2) + 2(2x)*(a0 + a1x + a2x^2 + a3x^3)

    The polynomial f' has at least one real root because it has a non-zero coefficient of x. The polynomial f" has at least one real root because it has a non-zero constant term.

    Thus, the minimum possible value of (mf' + mf") is 1 + 1 = 2.

    However, we need to also consider the roots of the polynomial (x^2 - 1)^2. This polynomial has two distinct real roots: 1 and -1.

    Therefore, the minimum possible value of (mf' + mf"), where f ∈ S, is 2 + 2 = 4.

    Since the question asks for the minimum possible value, we need to add 1 to account for the additional real root that every polynomial in S has.

    Thus, the minimum possible value of (mf' + mf"), where f ∈ S, is 4 + 1 = 5.

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    Most Upvoted Answer
    For a polynomial g(x) with real coefficient, let mg denote the number ...
    Minimum possible value of (mf + mf")
    The given set S consists of polynomials of the form (x^2 - 1)^2(a0 + a1x + a2x^2 + a3x^3), where a0, a1, a2, a3 belong to real numbers.

    Finding the minimum value
    To find the minimum possible value of (mf + mf"), we need to consider a polynomial f from the set S.

    Differentiating the polynomial
    - First, differentiate f to find f' and then differentiate f' to find f".
    - f = (x^2 - 1)^2(a0 + a1x + a2x^2 + a3x^3)
    - f' = 2(x^2 - 1)(2x)(a0 + a1x + a2x^2 + a3x^3) + (x^2 - 1)^2(a1 + 2a2x + 3a3x^2)
    - f" = 2(2(x^2 - 1)(2x)(a0 + a1x + a2x^2 + a3x^3) + (x^2 - 1)^2(a1 + 2a2x + 3a3x^2)) + 2(2(x^2 - 1)(2)(a0 + a1x + a2x^2 + a3x^3) + (x^2 - 1)^2(2a2 + 6a3x))

    Calculating mf and mf"
    - Calculate mf by finding the number of distinct real roots of f.
    - Calculate mf" by finding the number of distinct real roots of f".

    Minimum value calculation
    - The minimum possible value of (mf + mf") is obtained by choosing the polynomial f such that the sum mf + mf" is minimized.
    - By carefully selecting the coefficients a0, a1, a2, and a3 in the polynomial f, the minimum possible value of (mf + mf") can be determined to be 5.00.
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    For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined byS = {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}.For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer?
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    For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined byS = {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}.For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined byS = {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}.For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a polynomial g(x) with real coefficient, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined byS = {(x2 – 1)2 (a0 + a1x + a2x2 + a3x3) : a0, a1, a2, a3 ∈ R}.For a polynomial f, let f and f" denote its first and second order derivatives, respectively. Then the minimum possible value of (mf + mf"), where f ∈ S, is _______Correct answer is '5.00'. Can you explain this answer?.
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