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Question Number 24469 by Tinkutara last updated on 18/Nov/17

$$\mathrm{Let}2x+3y+4z=9,x,y,z>0\mathrm{then}$$$$\mathrm{the}\mathrm{maximum}\mathrm{value}\mathrm{of}{(1+x)}^2{(2+y)}^3$$$${(4+z)}^4\mathrm{is}$$

Commented by sushmitak last updated on 18/Nov/17

$$\mathrm{AM}>\mathrm{GM}$$$$\frac{(1+x)\times 2+(2+y)\times 3+(4+z)\times 4}{9}⩾{\left({(1+x)}^2{(2+y)}^3{(4+z)}^4\right)}^{1/9}$$$$\frac{24+9}{9}⩾{\left({(1+x)}^2{(2+y)}^3{(4+z)}^4\right)}^{1/9}$$$$\frac{11}{3}⩾{\left({(1+x)}^2{(2+y)}^3{(4+z)}^4\right)}^{1/9}$$$${\left(\frac{11}{3}\right)}^9⩾{(1+x)}^2{(2+y)}^3{(4+z)}^4$$

Commented by Tinkutara last updated on 18/Nov/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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