(B) Given $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ on $[-2, 2]$.Since $f(x)$ and $g(x)$ are polynomials,they are continuous on $[-2, 2]$ and differentia

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(B) Given $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ on $[-2, 2]$.Since $f(x)$ and $g(x)$ are polynomials,they are continuous on $[-2, 2]$ and differentiable on $(-2, 2)$. Thus,both satisfy the Mean Value Theorem.For Rolle's Theorem,we check $f(a) = f(b)$:$f(-2) = (-2)^{3} = -8$ and $f(2) = (2)^{3} = 8$. Since $f(-2) \neq f(2)$,$f(x)$ does not satisfy Rolle's Theorem.$g(-2) = (-2)^{3} - 4(-2) = -8 + 8 = 0$ and $g(2) = (2)^{3} - 4(2) = 8 - 8 = 0$. Since $g(-2) = g(2)$,$g(x)$ satisfies Rolle's Theorem.Therefore,statement $(a)$ and statement $(c)$ are correct.

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

Medium

If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:
$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.
$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.
$(c)$ Only $g(x)$ satisfies Rolle's Theorem.
Which of these statements is correct?

KCET 2014 All KCET

A
$(a)$ alone is correct
B
$(a)$ and $(c)$ are correct
C
$(a)$ and $(b)$ are correct
D
None is correct

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Continuity and Differentiation

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Rolle’s theorem, Lagrange's mean value theorem

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Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?

MediumAP EAMCET 2025

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MediumWBJEE 2019

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For the Mean Value Theorem $f(b) - f(a) = (b - a) f'(x_1)$ where $a < x_1 < b$,if $f(x) = 1/x$,then $x_1 = ?$

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If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.$(c)$ Only $g(x)$ satisfies Rolle's Theorem.Which of these statements is correct?

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Given $f(x) = 4 - (\frac{1}{2} - x)^{2/3}$,$g(x) = \begin{cases} \frac{\tan([x])}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases}$,$h(x) = \{x\}$,and $k(x) = 5^{\log_2(x + 3)}$. Then,in the interval $[0, 1]$,Lagrange's Mean Value Theorem is $NOT$ applicable to:

Consider the following functions:$I) f(x) = \begin{cases} \frac{1}{2}-x, & x < \frac{1}{2} \\ (\frac{1}{2}-x)^2, & x \geq \frac{1}{2} \end{cases}$$II) f(x) = |3x-1|$$III) f(x) = x|x|$$IV) f(x) = |x|$Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?

Let $f$ and $g$ be differentiable on the interval $I$ and let $a, b \in I, a < b$. Then,

Let $f(x)$ and $g(x)$ be two functions which are defined and differentiable for all $x \ge x_0$. If $f(x_0) = g(x_0)$ and $f'(x) > g'(x)$ for all $x > x_0$,then:

For the Mean Value Theorem $f(b) - f(a) = (b - a) f'(x_1)$ where $a < x_1 < b$,if $f(x) = 1/x$,then $x_1 = ?$

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If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:
$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.
$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.
$(c)$ Only $g(x)$ satisfies Rolle's Theorem.
Which of these statements is correct?