(A) Let $f(x) = (x^2-1)^2 h(x)$,where $h(x) = a_0+a_1x+a_2x^2+a_3x^3$.Since $(x^2-1)^2 = (x-1)^2(x+1)^2$,$f(x)$ has roots at $x=1$ and $x=-1$ with mul

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(A) Let $f(x) = (x^2-1)^2 h(x)$,where $h(x) = a_0+a_1x+a_2x^2+a_3x^3$.Since $(x^2-1)^2 = (x-1)^2(x+1)^2$,$f(x)$ has roots at $x=1$ and $x=-1$ with multiplicity at least $2$.Thus,$f(1)=0, f(-1)=0$ and $f'(1)=0, f'(-1)=0$.By Rolle's Theorem,there exists $\alpha \in (-1, 1)$ such that $f'(\alpha)=0$.Since $f'(x)$ has roots at $-1, \alpha, 1$,we have $m_{f'} \ge 3$.For $f(x) = (x^2-1)^2$,the roots are $1, -1$,so $m_f = 2$.Then $m_f + m_{f'} = 2 + 3 = 5$.Thus,the minimum value is $5$.

Class 12 Mathematics Continuity and Differentiability Rolle’s theorem, Lagrange's mean value theorem

For a polynomial $g(x)$ with real coefficients,let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{(x^2-1)^2(a_0+a_1x+a_2x^2+a_3x^3) : a_0, a_1, a_2, a_3 \in \mathbb{R}\}$. 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

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$5$
B
$8$
C
$9$
D
$10$

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Continuity and Differentiability

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Rolle’s theorem, Lagrange's mean value theorem

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For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0$, the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval

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If $f(x) = x^\alpha \log x$ and $f(0) = 0$,then the value of $\alpha$ for which Rolle's theorem can be applied in $[0, 1]$ is

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Examine the applicability of the Mean Value Theorem for the following functions:
$(i)$ $f(x) = [x]$ for $x \in [5, 9]$
$(ii)$ $f(x) = [x]$ for $x \in [-2, 2]$
$(iii)$ $f(x) = x^{2} - 1$ for $x \in [1, 2]$

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Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$. Then:

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Let $f$ and $g$ be twice differentiable even functions on $(-2, 2)$ such that $f(\frac{1}{4}) = 0, f(\frac{1}{2}) = 0, f(1) = 1$ and $g(\frac{3}{4}) = 0, g(1) = 2$. Then,the minimum number of solutions of $f(x)g''(x) + f'(x)g'(x) = 0$ in $(-2, 2)$ is equal to

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For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0$, the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval

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Examine the applicability of the Mean Value Theorem for the following functions:$(i)$ $f(x) = [x]$ for $x \in [5, 9]$$(ii)$ $f(x) = [x]$ for $x \in [-2, 2]$$(iii)$ $f(x) = x^{2} - 1$ for $x \in [1, 2]$

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Examine the applicability of the Mean Value Theorem for the following functions:
$(i)$ $f(x) = [x]$ for $x \in [5, 9]$
$(ii)$ $f(x) = [x]$ for $x \in [-2, 2]$
$(iii)$ $f(x) = x^{2} - 1$ for $x \in [1, 2]$