Question Number 134211 by rs4089 last updated on 01/Mar/21
Answered by mr W last updated on 01/Mar/21
![u=x^m y^n (a−x−y)^p (∂u/∂x)=y^n [mx^(m−1) (a−x−y)^p −x^m p(a−x−y)^(p−1) ] =y^n x^(m−1) (a−x−y)^(p−1) [m(a−x−y)−xp]=0 ⇒m(a−x−y)−xp=0 ...(i) similarly ⇒n(a−x−y)−yp=0 ...(ii) ⇒(x/y)=(m/n) ⇒x=((ma)/(m+n+p)) ⇒y=((na)/(m+n+p)) ⇒z=((pa)/(m+n+p)) u_(max) =(((ma)/(m+n+p)))^m (((na)/(m+n+p)))^n (((pa)/(m+n+p)))^p =((a/(m+n+p)))^(m+n+p) m^m n^n p^p](https://www.tinkutara.com/question/Q134228.png)
$${u}={x}^{{m}} {y}^{{n}} \left({a}−{x}−{y}\right)^{{p}} \\ $$$$\frac{\partial{u}}{\partial{x}}={y}^{{n}} \left[{mx}^{{m}−\mathrm{1}} \left({a}−{x}−{y}\right)^{{p}} −{x}^{{m}} {p}\left({a}−{x}−{y}\right)^{{p}−\mathrm{1}} \right] \\ $$$$={y}^{{n}} {x}^{{m}−\mathrm{1}} \left({a}−{x}−{y}\right)^{{p}−\mathrm{1}} \left[{m}\left({a}−{x}−{y}\right)−{xp}\right]=\mathrm{0} \\ $$$$\Rightarrow{m}\left({a}−{x}−{y}\right)−{xp}=\mathrm{0}\:\:…\left({i}\right) \\ $$$${similarly} \\ $$$$\Rightarrow{n}\left({a}−{x}−{y}\right)−{yp}=\mathrm{0}\:\:\:…\left({ii}\right) \\ $$$$\Rightarrow\frac{{x}}{{y}}=\frac{{m}}{{n}} \\ $$$$\Rightarrow{x}=\frac{{ma}}{{m}+{n}+{p}} \\ $$$$\Rightarrow{y}=\frac{{na}}{{m}+{n}+{p}} \\ $$$$\Rightarrow{z}=\frac{{pa}}{{m}+{n}+{p}} \\ $$$${u}_{{max}} =\left(\frac{{ma}}{{m}+{n}+{p}}\right)^{{m}} \left(\frac{{na}}{{m}+{n}+{p}}\right)^{{n}} \left(\frac{{pa}}{{m}+{n}+{p}}\overset{{p}} {\right)} \\ $$$$=\left(\frac{{a}}{{m}+{n}+{p}}\right)^{{m}+{n}+{p}} {m}^{{m}} {n}^{{n}} {p}^{{p}} \\ $$