Maximize $x^2+y^2+z^2$ subject to $x+2y+3z=4$ and $x,y,z\geq 0$

Question

If $x+2y+3z=4$, then find maximum value of $x^2+y^2+z^2$, where $x,y,z\geq 0$.

My try, using Lagrange multiplier method:

$$x^2+y^2+z^2=\lambda(x+2y+3z)$$

So $\displaystyle 2x=\lambda$ and $2y=2\lambda$ and $2z = 3\lambda$

Put $\displaystyle x=\frac{\lambda}{2}\;\;,y=z\;\;,z=\frac{3}{2}$ in $x+2y+3z=4$. We get $\displaystyle \lambda = \frac{4}{7}$

So $$\displaystyle (x^2+y^2+z^2)_{\max} = \frac{164}{49}\;,$$ When $\displaystyle x=\frac{2}{7}\;\;,y=\frac{4}{7}\;\;,z=\frac{12}{7}$

Is my solution right? If not then how can we calculate it?

Thanks

Answer 1

After replacing the constraint $x$, $y$, $z>0$ by $x$, $y$ , $z\geq0$ the feasible domain becomes a closed triangle $T$ with its vertices on the positive coordinate axes. Imagine a sphere centered at $O$, so large that it contains the triangle $T$ in its interior. Now continually shrink this sphere until it hits one of the vertices of $T$. This will be the vertex $X:=(4,0,0)$, since the other vertices are lying nearer to $O$. It follows that the point of $T$ farthest away from $O$ is $X$, and that the maximum of the given function on $T$ is $4^2=16$.

Answer 2

The choice of using Lagrange's multiplier is correct but your calculation seems a little correction i guess, you get $$2x=\lambda$$$$2y=2\lambda$$$$2z=3\lambda$$ using these values you put $$x=\lambda/2$$$$y=\lambda$$$$z=3\lambda/2$$ putting these values in the constraint function you get $\lambda=4/7$ using these values you get $$x=2/7$$$$y=4/7$$$$z=12/14$$

Answer 3

Your answer is clearly wrong. Put $y=z=0$ giving $x=4$, and $x^2+y^2+z^2=16$.

If $x+2y+3z=4$ then $x=4-2y-3z$. Substitute into $x^2+y^2+z^2$ giving \begin{equation*} 5y^2+10z^2+12yz-16y-24z+16 \end{equation*} as the function.

This has a MINIMUM value at $x=2/7, y=4/7, z=6/7$ using standard two-variable partial derivative methods. The maximum is unbounded.

Answer 4

The maximum of this function occurs at the boundary. i.e, at $x=4, y=0, z=0$ . The maximum happens to be $16$.

If you wanted the minimum though, It occurs at $x=\frac 27; y=\frac 47; z=\frac 67$ The minimum is $\frac {56}{49}$ . It follows your own way, with a few corrections.

Answer 5

Consider the product

$$ p=(x+2y+3z-4)(2x+y+6z+8) $$

Expanding $p$, we have

$$ p=2x^2+2y^2+18z^2+5xy+12xz+15yz+12y-32 \geq 2x^2+2y^2+18z^2-32 $$

Since $p=0$, dividing by $2$ we deduce

$$ x^2+y^2+z^2 \leq x^2+y^2+9z^2 \leq 16 $$

Equality is when $x=2,y=z=0$.