284 views

The maximum value of the function.

$f\left(x, y, z\right)= \left(x - 1 / 3\right)^{2}+ \left(y - 1 / 3\right)^{2}+ \left(z - 1 / 3\right)^{2}$

Subject to the constraints

$x + y + z=1, x \geq 0, y \geq 0, z \geq 0$

is

1. $1 / 3$
2. $2 / 3$
3. $1$
4. $4 / 3$
5. $4 / 9$
asked in Calculus | 284 views

Expanding given equation

$x^2 + y^2 + z^2 - \frac{2}{3}(x+y+z) + \frac{1}{3}$

$\quad = x^2 + y^2 + z^2 - \frac{2}{3} + \frac{1}{3}$

$\quad = (x+y+z)^2 - \frac{1}{3} - 2(xy + yz +xz)$

$\quad = 1 - \frac{1}{3} - 2(xy + yz +xz)$

$\quad = \frac{2}{3} - 2(xy + yz +xz)$

Now to maximize it, we need to minimize $(xy + yz + xz)$. As all $x,y$ and $z$ are non-negative $xy + yz + xz \geq 0.$ So, the maximum value is $\frac{2}{3}$.

Correct Option: B.
selected
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Hello dhruv

You answer and way to solve it , is absolutely correct.

Since all X,Y,Z are greater than equal to zero. So in X+Y+Z=1 means any two out of three variable will be Zero and remaining variable will have value one..

So, Max value will be

(1-1/3)2+(0-1/3)2+(0-1/3)2=2/3

+2
Ans is 4/9.
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How,?

I'm getting 16/9 after solving the above equation.
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@Shivam Dwivedi

I don't think so, did you consider the constraint x + y + z = 1? If you did, can you please tell how you got 4/9.

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+1 vote