Largest Inequality Bound
What is the largest integer N such that x 2 y 2 z 2 ( 1 − x 2 ) ( 1 − y 2 ) ( 1 − z 2 ) > N , as x , y and z range over all positive reals subject to x + y + z < 1 ?
The answer is 512.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
3 solutions
First of all, we know that to find N, we must find the smallest value of this function and then find a value that we know will be less than or equal to the smallest value. Next, let's analyze the equation. We can split the function into four pieces. ( 1 − x 2 ) , ( 1 − y 2 ) , ( 1 − z 2 ) , and ( x 2 y 2 z 2 1 ) . Looking at the first three, we can see that each piece will grow smaller as x,y and z grow bigger. The same can be said for the last piece. Therefore, to find the minimum value of this function, we must find the smallest values for x, y and z.
When we try out values for x,y and z, with the intentions of finding the largest values possible, we quickly see that when we make two of the variables larger, the third becomes smaller. Each variable's value affects the others. Therefore, to get the largest values for x, y and z, we must find values that add up to 1 and are all the same. So, if x, y and z are all the same value, the function will give its minimum value. x and y and z must all be less than 1/3 if x+y+z is less than 1. If we plug in x=1/3, y=1/3 and z=1/3 into the function, we get ( ( 1 − 1 / 9 ) 3 ) / ( ( 1 / 9 ) 3 ) . This simplifies to 512. 512 is smaller than any answer the function will give, as x, y and z cannot all be 1/3, as this would cause their sum to be equal to 1. Therefore, N = 5 1 2 .
not enough explanation. Incomplete solution.
@Sagnik Saha Does the updated version make sense to you?
@Mayankk Bhagat We are asked to find the largest value that is smaller than anything this function can produce. Basically, we must find the highest integer that is smaller than the minimum value of this function. To find that, we must follow the steps detailed above. Since x + y + z =1, instead of being less than 1, like the problem says. Therefore, 512 is a lower number than the function would ever produce, as the values of x, y and z are higher than they could be according to what the rules of this problem say. The higher the values for x, y and z, the smaller the value of this function will be. As a result, 512 is definitely the highest integer that exists that is smaller than any value of this function, provided that x+y+z< 1.
However, if the question asked for the maximum value of this function, we would get
l i m l i m l i m ( 1 − x 2 ) ( 1 − y 2 ) ( 1 − z 2 ) = ∞ x → 0 y → 0 z → 0 x 2 y 2 z 2
.
This function would tend towards infinity, as x, y and z get smaller, like you showed you in your example. I hope this answers your question. Does it make sense?
we have been asked to find the largest integer....... not the least!!!!!!! the highest value of this function can be 1000000000000000000000000000000000000000000000............................ since if we put x=y=z=0.0000000000000000000000000001 then still x+y+z<1
We use Karush-Kuhn-Tucker,basically it is just like lagrange multipliers except it can be applied in case of inequality constraints.
Solution outline:
We write the LHS as ( 1 + x ) ( 1 − x ) ( 1 + y ) ( 1 − y ) ( 1 + z ) ( 1 − z ) > N x 2 y 2 z 2 Next we substitute the value of (x+y+z) in place of 1 since x + y + z < 1 .
to obtain ( 1 + x ) ( 1 − x ) ( 1 + y ) ( 1 − y ) ( 1 + z ) ( 1 − z ) > ( 2 x + y + z ) ( y + z ) ( x + 2 y + z ) ( x + y ) ( x + y + 2 z ) ( x + y ) > N x 2 y 2 z 2
Now we apply AM -GM, ( x + y ) ≥ 2 x 1 / 2 y 1 / 2 ( 2 x + y + z ) = ( x + y + x + z ) ≥ 4 4 x 2 y z
Multiplying these equatione we get N = 5 1 2 .