As x,y,z ranges over all possible real numbers, what is the minimum value of 3x^2+12y^2+27z^2−4xy−12yz−6xz−8y−24z+100?
88,do tell if i am right
Is it 87
To find the minimum value of the expression 3x^2 + 12y^2 + 27z^2 - 4xy - 12yz - 6xz - 8y - 24z + 100, we can use a mathematical technique called completing the square.
First, let's rearrange the expression by grouping the like terms:
3x^2 - 4xy - 6xz + 12y^2 - 8y - 12yz + 27z^2 - 24z + 100
Now, let's focus on the terms involving x. We want to write them in the form (x - h)^2, where h is a constant. To do this, we need to complete the square.
For the terms involving x, we will ignore the coefficient 3 for now and only focus on the x terms: 3x^2 - 4xy - 6xz. To complete the square, we need to find a constant (let's call it a) such that (x - a)^2 gives 3x^2 - 4xy - 6xz when expanded.
To find this constant, we take the coefficient of x (which is -4y - 6z) and divide it by 2. In this case, a = (-4y - 6z) / 2.
Now, let's add and subtract the square of this constant within the expression to complete the square:
3x^2 - 4xy - 6xz + a^2 - a^2 + 12y^2 - 8y - 12yz + 27z^2 - 24z + 100
Looking at the x terms, we have:
(x - a)^2 + 12y^2 - 8y - 12yz + 27z^2 - 24z + 100 - a^2
Now, let's do the same with the y terms. We want to write them in the form (y - k)^2. To find the constant k, we divide the coefficient of y (which is -12z - 8) by 2. So k = (-12z - 8) / 2.
Adding and subtracting the square of this constant within the expression, we have:
(x - a)^2 + 12(y - k)^2 + 27z^2 - 24z + 100 - a^2 - k^2
Finally, let's focus on the z terms. We want to write them in the form (z - m)^2, so we divide the coefficient of z (which is -24) by 2. This gives us m = -24 / 2 = -12.
Adding and subtracting the square of this constant within the expression, we get:
(x - a)^2 + 12(y - k)^2 + 27(z - m)^2 + 100 - a^2 - k^2 - m^2
Simplifying this expression, we have:
(x - a)^2 + 12(y - k)^2 + 27(z - m)^2 + 100 - a^2 - k^2 - m^2
Now, we can see that the minimum value of this expression occurs when all the squared terms are equal to zero. This happens when x - a = 0, y - k = 0, and z - m = 0.
Therefore, the minimum value of the given expression is simply the constant term:
100 - a^2 - k^2 - m^2
Substituting the values of a, k, and m that we found earlier, we can calculate the minimum value.
Note: The values of a, k, and m depend on the coefficients of the original expression, in this case, -4y - 6z, -12z - 8, and -24, respectively.