Question

If x+y=8, then the maximum value of xy is

(a) 8

(b) 16

(c) 20

(d) 24

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)16 \\ \mathrm{Given}:x+y=8 \\ \Rightarrow y=8-x...\left(1\right) \\ \mathrm{Let}f\left(x\right)\mathrm{be}xy. \\ \Rightarrow f\left(x\right)=x\left(8-x\right)\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right] \\ \Rightarrow f\text{'}\left(x\right)=8-2x \\ \mathrm{For}\mathrm{a}\mathrm{local}\mathrm{maxima}\mathrm{or}\mathrm{a}\mathrm{local}\mathrm{minima},\mathrm{we}\mathrm{must}\mathrm{have} \\ f\text{'}\left(x\right)=0 \\ \Rightarrow 8-2x=0 \\ \Rightarrow 8=2x \\ \Rightarrow x=4 \\ \Rightarrow y=8-4=4\left[\mathrm{From}\mathrm{eq}.\left(1\right)\right] \\ \mathrm{Now}, \\ f\text{'}\text{'}\left(x\right)=-2 \\ \Rightarrow f\text{'}\text{'}\left(4\right)=-2<0 \\ \mathrm{So},x=4\mathrm{is}\mathrm{a}\mathrm{local}\mathrm{maxima}. \\ \mathrm{Hence},\mathrm{the}\mathrm{local}\text{maximum}\mathrm{value}\mathrm{is}\mathrm{given}\mathrm{by} \\ f\left(4\right)=4\times 4=16 \end{gathered}$$