Question

JEE ADVANCED MATHS QUESTION SEPTEMBER 2020

Answer 1

For a polynomial g(x) with real coefficient, let m_g denotes the number of distinct real roots of g(x), suppose S is the set of polynomial 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 (m_f' + m_f") where f ∈ S , is ...

solution : let h(x) = (a₀ + a₁x + a₂x² +a₃x³)

so, f(x) = (x² - 1)² h(x)

at x = 1, f(1) = (1² - 1)² h(1) = 0

at x = -1, f(-1) = {(-1)² - 1}²h(1) = 0

by Roll's theorem, there must be a point between -1 to 1 where f'(x) = 0,

let α ∈ (-1, 1) where f'(α) = 0

f'(x) = 2x(x² - 1)h(x) + (x² - 1)² h'(x)

here we also see , f'(1) = f'(-1) = 0

so f'(x) = 0 has atleast three real roots -1, α, 1 with -1 < α < 1

so, minimum possible value of mf' = 3

Now by Roll's theorem at least one real roots lies between (-1, α) and at least one real root lies between (α, 1) of f"(x)

so minimum value of mf" = 2

Therefore the minimum possible value of mf' + mf" = 3 + 2 = 5