Column - I gives functions which satisfy conditions of CMVT an specified interval and Column - II gi

Question

Column - I gives functions which satisfy conditions of CMVT an specified interval and Column - II gives value of ' $C$ ' for which LMVT is satisfied

Professor PC Thomas Classes · Answer

$f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$ for LMVT
$f^{\prime}(c)=\frac{0+1}{3}$ for 'a' $\Rightarrow2c-2=\frac{1}{3}\Rightarrow2c=\frac{7}{3}\Rightarrow c=\frac{7}{6}$
$f^{\prime}(c)=\frac{1}{3}$ for 'b' $\Rightarrow2-2c=\frac{1}{3}\Rightarrow2c=\frac{5}{3}\Rightarrow c=\frac{5}{6}$
$f^{\prime}(c)=\frac{1-3}{1+1}\Rightarrow3c^{2}-4c-1=-1 \Rightarrow 3c^{2}-4c=0 \Rightarrow c(3c-4)=0 \Rightarrow c=\frac{4}{3}$ (wait, $c \in [-1,1]$)
$f^{\prime}(c)=\frac{6}{3}\Rightarrow3c^{2}-12c+11=2 \Rightarrow 3c^{2}-12c+9=0 \Rightarrow c^{2}-4c+3=0, c=1,3 	herefore c=3 \in [1,4]$