# If $f$ and $g$ are differentiable functions in $\left[ {0,1} \right]$ satisfying $f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) = 0$

And $f\left( 1 \right) = 6$, then for some $c \ in \left[ {0,1} \right]$

$

{\text{A}}{\text{. 2}}f'\left( c \right) = g'\left( c \right) \\

{\text{B}}{\text{. }}2f'\left( c \right) = 3g'\left( c \right) \\

{\text{C}}{\text{. }}f'\left( c \right) = g'\left( c \right) \\

{\text{D}}{\text{. }}f'\left( c \right) = 2g'\left( c \right) \\

$

Last updated date: 18th Mar 2023

•

Total views: 306k

•

Views today: 4.85k

Answer

Verified

306k+ views

Hint: - Here two differentiable functions are given and their differentiability range is also given so use Lagrange’s mean value theorem (LMVT) for further solution. And proceed further from what is given in question.

We have given:

$f\left( 0 \right) = 2 = g\left( 1 \right)$,$g\left( 0 \right) = 0$ and $f\left( 1 \right) = 6$

As given in question both $f$ and $g$ are differentiable in $\left[ {0,1} \right]$

From LMVT

$f'\left( c \right) = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}$ here $\left( {b = 1,a = 0} \right)$$$$$

So, $f'\left( c \right) = \left( {\dfrac{{f\left( 1 \right) - f\left( 0 \right)}}{{1 - 0}}} \right)$$ = \dfrac{{6 - 2}}{1}$

$\therefore f'\left( c \right) = 4$.$ \ldots \ldots \left( 1 \right)$

Now apply LMVT on function $g$.

Now $g'\left( c \right) = \dfrac{{g\left( b \right) - g\left( a \right)}}{{b - a}}\left( {b = 1,a = 0} \right)$

$g'\left( c \right) = \dfrac{{g\left( 1 \right) - g\left( 0 \right)}}{{1 - 0}} = \dfrac{{2 - 0}}{1} = 2$$ \ldots \ldots \left( 2 \right)$

From equation $\left( 1 \right)$and $\left( 2 \right)$

$f'\left( c \right) = 2g'\left( c \right)$

Hence the option ${\text{D}}$ is the correct option.

Note: -Whenever you get this type of question the key concept of solving is you have to understand from the question that by using LMVT you can proceed further. You have to consider the range in which you have to apply LMVT.

We have given:

$f\left( 0 \right) = 2 = g\left( 1 \right)$,$g\left( 0 \right) = 0$ and $f\left( 1 \right) = 6$

As given in question both $f$ and $g$ are differentiable in $\left[ {0,1} \right]$

From LMVT

$f'\left( c \right) = \dfrac{{f\left( b \right) - f\left( a \right)}}{{b - a}}$ here $\left( {b = 1,a = 0} \right)$$$$$

So, $f'\left( c \right) = \left( {\dfrac{{f\left( 1 \right) - f\left( 0 \right)}}{{1 - 0}}} \right)$$ = \dfrac{{6 - 2}}{1}$

$\therefore f'\left( c \right) = 4$.$ \ldots \ldots \left( 1 \right)$

Now apply LMVT on function $g$.

Now $g'\left( c \right) = \dfrac{{g\left( b \right) - g\left( a \right)}}{{b - a}}\left( {b = 1,a = 0} \right)$

$g'\left( c \right) = \dfrac{{g\left( 1 \right) - g\left( 0 \right)}}{{1 - 0}} = \dfrac{{2 - 0}}{1} = 2$$ \ldots \ldots \left( 2 \right)$

From equation $\left( 1 \right)$and $\left( 2 \right)$

$f'\left( c \right) = 2g'\left( c \right)$

Hence the option ${\text{D}}$ is the correct option.

Note: -Whenever you get this type of question the key concept of solving is you have to understand from the question that by using LMVT you can proceed further. You have to consider the range in which you have to apply LMVT.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?

Alfred Wallace worked in A Galapagos Island B Australian class 12 biology CBSE

Imagine an atom made up of a proton and a hypothetical class 12 chemistry CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

How do you define least count for Vernier Calipers class 12 physics CBSE

Why is the cell called the structural and functional class 12 biology CBSE

A 30 solution of H2O2 is marketed as 100 volume hydrogen class 11 chemistry JEE_Main