21''Equality holds when all terms are equal...."
This requires some additional constraints like their product is constant ..
m i correct ?
x+y+z=pi (angles of a triangle)
show that sin x + sin y + sin z >= 3√3/2
Solution
\frac{\sin x+\sin y+\sin z}{3}\ge(sin x\times\sin y\times\sin z)^{1/3} (AM GM Inequality)
Equality holds when all terms are equal....
So sin x + sin y + sin z >= 3sin (pi/3)>=3√3/2
Hence Proved...
Find the flaw in the solution
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17 Answers
49oops.. sorry..
I was wrong..
sin x , sin y , sin z cannot be negative! [3]
21'' but not the argument that follows ''
why ??
can u explain more exactly ?
1Hint - The R . H . S of the AM - GM inequality achieves its maximal value at the exact point at which the L . H .S is maximum . So , the inequality is valid , but not the argument that follows .
21Chinmaya..u r again wrong.. it's not possible to have any angle greater than 180
read the qstn properly.. it says angles of a triangle
1since we are given sum of angles is pi...it is possible to have one of them greater than 180°,and that would render sine of that angle as a negative quantity....from which AM-GM fails!
62Chinmaya.. lets use ur logic..
Sin x = sin y = sin z
so from there even if it happens, x y and z are in 0 to pi
so we have atleast one of the two as equal...
the third being pi-2x
so we have
sin x + sin x + sin pi-2x
= 2 sin x + sin 2x
Even this step would have been wrong...
1i don't find anywhere that it is given that one of the angles can't be concave like(i.e. greater than 180°)[3]...sorry for this horrible term i used
211sorry if i wasn't clear enough in my first reply....actually i thought saying that proof is based on sin x =sin y=sin z (with x+y+z=pi) and not x=y=z was sufficient to include this positivity thing as well...maybe i need to learn how to write my reasoning properly[1]..else i would continue making a mess like this!
1dude my argument is much wider than most might have realized...AM GM holds for positive entitites only...so,it can't be taken as a proof.
here's an analogy:
dx^n/dx =nx^n-1
proving it for n=real numbers is much different from just proving it for n=natural numbers...hope i am clear engouh now shubhomoy
1doesn't ur proof say that sin x =sin y=sin z (with x+y+z=pi) and not x=y=z.
21moreover when all the terms are equal then just AM = Gm
that doesn't prove that inequality at all .. does it?