For a polynomial g(x) with real coefficient, let $m_g$ denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by $$S=\left\{\left(x^{2}-1\right)^{2}\left(a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\right): a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb{R}\right\}.$$ For a polynomial f, let f' and f" denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'}+m_{f''}\right)$, where $f \in S$, is ____

JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · Application of Derivatives

For a polynomial g(x) with real coefficient, let

m_{g}

denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by

S = {(x^{2} - 1)^{2} (a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}) : a_{0}, a_{1}, a_{2}, a_{3} \in 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 \in S

, is ____

Answer: 5

Let

f (x) = (x^{2} - 1)^{2} h (x)

where

h (x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3}

.

Since

f (1) = f (- 1) = 0

, by Rolle's theorem,

f^{'} (α) = 0

for some

α \in (- 1, 1)

.

Also,

f^{'} (1) = f^{'} (- 1) = 0

(by direct computation or noting the double roots).

So

f^{'} (x) = 0

has at least 3 roots:

- 1, α, 1

with

- 1 < α < 1

.

By Rolle's theorem applied to

f^{'}

f^{''} (x) = 0

has at least 2 roots, say

β, γ

with

- 1 < β < α < γ < 1

.

Thus

min (m_{f^{''}}) = 2

.

For

f (x) = (x^{2} - 1)^{2}

(i.e.,

h (x) = 1

$f^{'} (x) = 4 x (x^{2} - 1) = 4 x (x - 1) (x + 1)$ has roots $- 1, 0, 1$ , so $m_{f^{'}} = 3$ .
$f^{''} (x) = 12 x^{2} - 4$ has roots $\pm \frac{1}{3}$ , so $m_{f^{''}} = 2$ .

Therefore,

m_{f^{'}} + m_{f^{''}} = 3 + 2 = 5

JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · Numerical answer