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CBSE
Mathematics
Lagrange's mean value theorem
How do you prove that if a group G has no non-trivial subgroups, then it is finite of prime order?
CBSE
Mathematics
Lagrange's mean value theorem
How small can $f(4)$ possibly be? If $f$ is a function such that $f(1){\text{ }} = {\text{ }}4$ and $f'(x){\text{ }} \geqslant {\text{ }}2$ for $1{\text{ }} \leqslant {\text{ }}x{\text{ }} \leqslant {\text{ }}4$ respectively.
(a) $11$
(b) $16$
(c) $10$
(d) $15$

CBSE
Mathematics
Lagrange's mean value theorem
For the function $ f\left( x \right) = {e^x} $ , $ a = 0 $ and $ b = 1 $ , the value of c in mean value theorem will be
(A) $ \log x $
(B) $ \log \left( {e - 1} \right) $
(C) $ 0 $
(D) $ 1 $

CBSE
Mathematics
Lagrange's mean value theorem
Verify the hypothesis and conclusion of Lagrange’s mean value theorem for the function \[f\left( x \right) = \dfrac{1}{{4x - 1}},\, 1 \leqslant x \leqslant 4\].
CBSE
Mathematics
Lagrange's mean value theorem
The value of $c$ from the Lagrange’s mean value theorem for which $f(x) = \sqrt {25 - {x^2}} $ in $[1,5]$ is
A) $5$
B) $1$
C) $\sqrt {15} $
D) None of these
CBSE
Mathematics
Lagrange's mean value theorem
Examine the application of Mean Value Theorem for all three functions given.
\[f\left( x \right)=\left[ x \right]forx\in \left[ -2,2 \right]\]
CBSE
Mathematics
Lagrange's mean value theorem
Check the applicability of Lagrange’s mean value theorem for $f(x) = \sqrt {5 - x} $ on $\left[ { - 3,6} \right]$.
CBSE
Mathematics
Lagrange's mean value theorem
How do I find the numbers $c$ that satisfies the Mean Value Theorem for
 \[f(x) = {x^3} + x - 1\] on the interval \[[0,3] \] ?

CBSE
Mathematics
Lagrange's mean value theorem
Explain how one can prove Bernoulli's inequality with the help of Mean Value Theorem?
CBSE
Mathematics
Lagrange's mean value theorem
How do you determine the values of c that satisfy the mean value theorem on the interval \[\left[ {\dfrac{\pi }{2},\dfrac{{3\pi }}{2}} \right] \] for \[f\left( x \right) = \sin \left( {\dfrac{x}{2}} \right)\] ?
CBSE
Mathematics
Lagrange's mean value theorem
How do you find the value of $ c $ that satisfy the equation $ \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}} = f'\left( c \right) $ in the conclusion of the mean value theorem for the function $ f\left( x \right) = 4{x^2} + 4x - 3 $ on the interval $ \left[ { - 1,0} \right] $ ?
CBSE
Mathematics
Lagrange's mean value theorem
For what values of $a,m$ and $b$ Lagrange’s mean value theorem is applicable to the function $f\left( x \right)$ for $x \in \left[ {0,2} \right]$
$f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  3&{x = 0}&{} \\
  { - {x^2} + a}&{0 < x < 1}&{} \\
  {mx + b}&{1 \leqslant x \leqslant 2}&{}
\end{array}} \right.$
I. $a = 3;m = 2;b = 0$
II. $a = 3;m = - 2;b = 4$
III. $a = 3;m = 2;b = 1$
IV. No such $a,m,b$ exist
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