Let x,y,z be positive real numbers such that:
xyz=945
x(y+1)+y(z+1)+z(x+1)=385.
What is the minimum possible value of z+y/2+x/4?
To find the minimum possible value of z + y/2 + x/4, we first need to simplify the given expressions and then apply the appropriate mathematical techniques. Let's go step by step:
1. Given: xyz = 945
We are given that the product of x, y, and z is 945.
2. Given: x(y+1) + y(z+1) + z(x+1) = 385
Expand the expression:
xy + x + yz + y + zx + z = 385
3. Simplify the equation:
xy + yz + zx + x + y + z = 385
4. Rearrange the equation:
x(y + z + 1) + y(z + 1) + z = 385
(x + y + z) + (xy + x + yz + y + zx + z) = 385
x + y + z + xy + yz + zx + 2(y + z) + 1 = 385
x + y + z + xy + yz + zx + 2(y + z) = 384
5. Substitute xyz = 945
If we subtitute xyz into the equation:
x + y + z + xy + yz + zx + 2(y + z) = 384
x + y + z + xy + yz + zx + 2(y + z) = 384
x + y + z + xy + yz + zx + 2(y + z) = (xyz) - 561
x + y + z + xy + yz + zx + 2(y + z) = 945 - 561
x + y + z + xy + yz + zx + 2(y + z) = 384
6. Simplify the above equation:
xy + yz + zx + x + y + z + 2y + 2z = 384
xy + yz + zx + x + y + z + 2(y + z) = 384
7. Substitute z + y/2 + x/4 into the equation:
If we substitute the given expression z + y/2 + x/4 into the equation, we get:
xy + yz + zx + x + y + z + 2(y + z) = 384
xy + yz + zx + x + y + z + 2(y + z) = 384
xy + yz + zx + x + y + z + 2(z + y/2 + x/4) = 384
xy + yz + zx + x + y + z + 2z + y + x/2 = 384
xy + yz + zx + x + y + z + (4z + 2y + x)/2 = 384
8. Simplify and solve for the minimum value:
Since we want to find the minimum value of z + y/2 + x/4, we need to minimize the above expression.
For this, we can look at the coefficient of each variable (xy, yz, zx, x, y, z) and find the corresponding value of the variable that minimizes the expression.
By observation, since all the variables are positive real numbers, we see that the minimum value of xy, yz, zx, x, y, and z is zero. When any of them is zero, it contributes the least to the expression.
Therefore, we can conclude that the minimum value of z + y/2 + x/4 is when xy = yz = zx = x = y = z = 0.
Thus, the minimum possible value of z + y/2 + x/4 is 0.