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mars1129 [50]
3 years ago
11

Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Mathematics
1 answer:
Allushta [10]3 years ago
5 0
Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
f_x=5yz=0\implies y=0\text{ or }z=0
f_y=5xz=0\implies x=0\text{ or }z=0
f_z=5xy=0\implies x=0\text{ or }y=0


Taken together, we find that (0, 0, 0) appears to be the only critical point on f within the ball. At this point, we have f(0,0,0)=0.

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)

with partial derivatives (set to 0)

L_x=5yz+2\lambda x=0
L_y=5xz+2\lambda y=0
L_z=5xy+2\lambda z=0
L_\lambda=x^2+y^2+z^2-1=0

We then observe that

xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2

So, ignoring the critical point we've already found at (0, 0, 0),


5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}
5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}
5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of \left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right), at which points we get a value of either of \pm\dfrac5{\sqrt3}, with the maximum being the positive value and the minimum being the negative one.
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Answer:

Cuando María afirma que si unen sus dos cuartos de cartulina obtendrán el medio pliego que necesitan, esto es:

  • <u>Verdadero</u>.

Step-by-step explanation:

Para entender mejor el ejercicio vamos a utilizar números cada vez que se habla de cantidades de cartulina, por lo tanto, María y Juan tienen 1/4 de cartulina cada uno, es decir, 1/4 * 2, y necesitan 1/2 pliego para poder realizar su tarea, por lo tanto, con la afirmación de María sobre unir los dos cuartos de cartulina, en caso de que sea verdadero, ocurrirá que la suma de los dos cuartos dará el medio pliego, como se muestra a continuación:

  • Total de cartulina de María y Juan = \frac{1}{4}+\frac{1}{4}
  • Total de cartulina de María y Juan = \frac{4+4}{16}
  • Total de cartulina de María y Juan = \frac{8}{16}

Procedemos a simplificar el fraccionario obtenido, sacando mitad tanto en el numerador como en el denominador:

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Como puedes ver al final, <u>la cantidad de cartulina de ambos, al ser sumada, da como resultado el 1/2 (medio) pliego que necesitan para su tarea de sociales, por lo cual la afirmación de María es correcta</u>.

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 <span>To minimize the perimeter you should always have a square. 
sqrt(289) = 17 
The dimensions should be 17 X 17 

To see , try starting at length 1, and gradually increase the length. 
The height decreases at a faster rate than the length increases, up until you reach a square. 

Or if you want to use algebra, Say the width is 17-x 
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Now, this is bigger than 17+x, as shown here: 
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Again, the dimensions should be a square. 17 X 17.</span>
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