Observe the following lists
List - I                                                  List - II
A) If x + y = k then $x^2 + y^2$            1)125
is minimum if
B) If x + y = 20 then $x^2{y}^3$                 2) x = y
is maximum when
C) the minimum value of                  3) x = 8
$64 sec\theta +27\cos ec\theta$ when and y = 12 $\theta \in (0,\pi /2)$
D) the greatest value of $\frac{\log x}{x}$        4) e
                                                               5) 1/e

$\left(\mathrm{A}\right) \mathrm{x}+\mathrm{y}=\mathrm{K} \mathrm{and} {\mathrm{x}}^2+{\mathrm{y}}^2 \mathrm{is} \mathrm{minimum} \Rightarrow \mathrm{x}=\mathrm{y}$

$$\begin{gathered} \left(\mathrm{B}\right) \mathrm{x}+\mathrm{y}=20 \\ \Rightarrow \mathrm{y}=20-\mathrm{x} \\ \mathrm{Let} \mathrm{S}={\mathrm{x}}^2{\mathrm{y}}^3={\mathrm{x}}^2{\left(20-\mathrm{x}\right)}^3 \\ \log \mathrm{s}=2 \log \mathrm{x}+3 \log \left(20-\mathrm{x}\right) \end{gathered}$$

$$\begin{gathered} \frac{1}{\mathrm{S}}{\mathrm{S}}^{\text{'}}=\frac{2}{\mathrm{x}}+\frac{3}{20-\mathrm{x}}\left(-1\right) \\ \mathrm{S} \mathrm{is} \min \mathrm{or} \max \\ \Rightarrow {\mathrm{S}}^{\text{'}}=0 \end{gathered}$$

$$\begin{gathered} \frac{2}{x}-\frac{3}{20-x}=0 \\ \Rightarrow 40-2x-3x=0 \\ \Rightarrow 40=5x\Rightarrow x=8 \\ x=8\Rightarrow y=20-x \\ =20-8 \\ =12 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{C}\right) \\ \mathrm{Let} \mathrm{S}=64\sec \mathrm{\theta }+27 \mathrm{cosec} \mathrm{\theta } \\ {\mathrm{S}}^{\text{'}}=64 \sec \mathrm{\theta } \tan \mathrm{\theta }-27 \mathrm{cosec} \mathrm{\theta } \cot \mathrm{\theta } \end{gathered}$$

S is min or max 

$$\begin{gathered} \Rightarrow S^{\text{'}}=0 \\ 64 \frac{\sin \theta }{{\cos }^2\theta }=\frac{27 \cos \theta }{{\sin }^2\theta } \\ 64 {\sin }^3\theta =27 {\cos }^3\theta \\ {\tan }^3\theta =\frac{27}{64} \\ \tan \theta =\frac{3}{4} \end{gathered}$$

$$\begin{gathered} \sec \mathrm{\theta }=\frac{5}{4} \mathrm{cosec} \mathrm{\theta }=\frac{5}{3} \\ \mathrm{S}=64\times \frac{5}{4}+27\left(\frac{5}{3}\right) \\ =80+45=125 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{D}\right) \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{\log \mathrm{x}}{\mathrm{x}} \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=\frac{\mathrm{x}.{\displaystyle \frac{1}{\mathrm{x}}}-\log \mathrm{x}.\left(1\right)}{{\mathrm{x}}^2} \\ =\frac{1-\log \mathrm{x}}{{\mathrm{x}}^2} \end{gathered}$$

$\mathrm{f}\left(\mathrm{x}\right)$ is min or max

$$\begin{gathered} \Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=0 \\ 1-\log \mathrm{x}=0 \\ \log \mathrm{x}=1 \\ \mathrm{x}=\mathrm{e} \\ \mathrm{greast} \mathrm{value} \mathrm{is} \\ =\mathrm{f}\left(\mathrm{e}\right)=\frac{\log \mathrm{e}}{\mathrm{e}}=\frac{1}{\mathrm{e}} \end{gathered}$$