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Find the maximum value of xyz when `(x)/(1)+(y^2)/(4)+(z^3)/(27)=1`, where `x,y,zgt 0`.

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To find the maximum value of \( xyz \) given the constraint \[ \frac{x}{1} + \frac{y^2}{4} + \frac{z^3}{27} = 1 \] where \( x, y, z > 0 \), we can use the method of inequalities, specifically the Arithmetic Mean-Geometric Mean (AM-GM) inequality. ...
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CENGAGE ENGLISH-INEQUALITIES INVOLVING MEANS -Illustration
  1. about to only mathematics

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  2. If a+b+c=1, then prove that 8/(27a b c)>{1/a-1}{1/b-1}{1/c-1}> 8.

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  3. If y z+z x+x y=12 ,w h e r ex ,y ,z are positive values, find the grea...

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  4. If a ,b ,c are positive, then prove that a//(b+c)+b//(c+a)+c//(a+b)geq...

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  5. Prove that 2^n >1+nsqrt(2^(n-1)),AAn >2 where n is a positive integer.

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  6. If S+a1+a2+a3++an ,a1 in R^+ for i=1ton , then prove that S/(S-a1)+S/...

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  7. If a1+a2+a3+......+an=1 AA ai > 0, i=1,2,3,......,n, then find the ma...

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  8. If a , b , c , are positive real numbers, then prove that (2004, 4M) {...

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  9. Prove that (sec^4alpha)/(tan^2beta)+(sec^4beta)/(tan^2alpha)ge8. If ea...

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  10. Prove that [(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(...

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  11. Prove that 1^1xx2^2xx3^3xxxxu nlt=[(2n+1)//3]n(n+1)//2,n in Ndot

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  12. Find the greatest value of x^2 y^3, where x and y lie in the first qua...

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  13. Find the maximum value of (7-x)^4(2+x)^5w h e nx lies between -2a n d7...

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  14. Find the maximum value of xyz when (x)/(1)+(y^2)/(4)+(z^3)/(27)=1, whe...

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  15. If a ,b >0 such that a^3+b^3=2, then show that a+blt=2.

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  16. If m >1,n in N show that 1m+2m+2^(2m)+2^(3m)++2^(n m-m)> n^(i-m)(2^n-...

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  17. Prove that in an acute angled triangle ABC , sec A+sec B +sec Cge 6.

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  18. Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)> a+b+c

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  19. Prove that (a^8+b^8+c^8)/(a^3b^3c^3)>1/a+1/b+1/c

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  20. If a ,b ,a n dc are positive and a+b+c=6, show that (a+1//b)2+(b+1//c)...

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