(A) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x$.We are given that $f(0) = 0$ and $f(\alpha) = 0$.Since $f(x)$ is a polynomial,it is continuous

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(A) Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x$.We are given that $f(0) = 0$ and $f(\alpha) = 0$.Since $f(x)$ is a polynomial,it is continuous on $[0, \alpha]$ and differentiable on $(0, \alpha)$.By Rolle's Theorem,there exists at least one point $c \in (0, \alpha)$ such that $f'(c) = 0$.Note that $f'(x) = na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1$.Therefore,the equation $f'(x) = 0$ has at least one positive root in the interval $(0, \alpha)$,which means the root is less than $\alpha$.

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

Medium

If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?

A
Less than $\alpha$
B
Greater than $\alpha$
C
Equal to $\alpha$
D
Greater than or equal to $\alpha$

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Class 12

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Chapter

Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

Let $f(x)=2+\cos x$ for all real $x$.
$STATEMENT-1$: For each real $t$,there exists a point $c$ in $[t, t+\pi]$ such that $f^{\prime}(c)=0$. because
$STATEMENT-2$: $f(t)=f(t+2\pi)$ for each real $t$.

DifficultIIT 2007

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Consider all functions given in List-$I$ in the interval $[1,3]$. List-$II$ has the values of '$c$' obtained by applying Lagrange's Mean Value Theorem $(LMVT)$ on the functions of List-$I$. Match the functions and values of '$c$'.

List-$I$ List-$II$

$A. |x-1|$ $I. 2 \log (e^3+e^2)$

$B. \log x$ $II. 2$

$C. x^2+x+1$ $III. \log_3 e^2$

$D. e^x$ $IV. \sqrt{2}$

$V. \log \left(\frac{e^3-e}{2}\right)$

List-$I$	List-$II$
$A. \|x-1\|$	$I. 2 \log (e^3+e^2)$
$B. \log x$	$II. 2$
$C. x^2+x+1$	$III. \log_3 e^2$
$D. e^x$	$IV. \sqrt{2}$
	$V. \log \left(\frac{e^3-e}{2}\right)$

DifficultTS EAMCET 2025

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$f:[1,3] \rightarrow R$ is a function defined as $f(x)=x^3+a x^2+b x$. If $f(1)-f(3)=0$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $a-b$ is equal to

MediumTS EAMCET 2021

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Let $a, b, c$ be real numbers such that $2a + 3b + 6c = 0$ and $g(x) = ax^2 + bx + c = 0$ has at least one root in the interval $(1, 2)$. If a function $f: [1, 2] \rightarrow \mathbb{R}$ for which Rolle's Theorem holds is such that $f(x)$ is a primitive of $g(x)$,then $f(x) = $

MediumAP EAMCET 2019

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Verify the Mean Value Theorem for the function $f(x) = x^{2} - 4x - 3$ in the interval $[a, b]$,where $a = 1$ and $b = 4$.

Medium

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If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?

Similar Questions

Let $f(x)=2+\cos x$ for all real $x$.$STATEMENT-1$: For each real $t$,there exists a point $c$ in $[t, t+\pi]$ such that $f^{\prime}(c)=0$. because$STATEMENT-2$: $f(t)=f(t+2\pi)$ for each real $t$.

$f:[1,3] \rightarrow R$ is a function defined as $f(x)=x^3+a x^2+b x$. If $f(1)-f(3)=0$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $a-b$ is equal to

Let $a, b, c$ be real numbers such that $2a + 3b + 6c = 0$ and $g(x) = ax^2 + bx + c = 0$ has at least one root in the interval $(1, 2)$. If a function $f: [1, 2] \rightarrow \mathbb{R}$ for which Rolle's Theorem holds is such that $f(x)$ is a primitive of $g(x)$,then $f(x) = $

Verify the Mean Value Theorem for the function $f(x) = x^{2} - 4x - 3$ in the interval $[a, b]$,where $a = 1$ and $b = 4$.

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Let $f(x)=2+\cos x$ for all real $x$.
$STATEMENT-1$: For each real $t$,there exists a point $c$ in $[t, t+\pi]$ such that $f^{\prime}(c)=0$. because
$STATEMENT-2$: $f(t)=f(t+2\pi)$ for each real $t$.