Verified Solution • 2025–26 Session moderate

QJEEAdv20Ma9a87: JEE Advanced 2020 Mathematics - Class 11 JEEAdv20Ma9a87 Solution: For a polynomial g(x) with real coefficients, let m g denote…

For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients 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 ..............

Step-by-Step Verified Solution

Given set S of polynomials with real coefficients

S={(x2−1)2(a0+a1x+a2x2+a3x3):a0,a1,a2,a3∈R}

and for a polynomial f∈S, Let

f(x)=(x2−1)2(a0+a1x+a2x2+a3x3)

it have −1 and 1 as repeated roots twice, so graph of f(x) touches the X-axis at x = −1 and x = 1, so f'(x) having at least three roots x = −1, 1 and α. Where α∈(−1, 1) and f''(x) having at least two roots in interval (−1, 1)

So, mf' = 3 and mf'' = 2

∴ Minimum possible value of (mf' + mf'') = 5

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