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- Aug 24th 2007, 11:56 AM #1

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## Max / Min

Hello

I had following exercise at my exam last wednesday:

Find the nominal value and the place of the maxmum and the minimum of the function f(x,y,z) = x + y + z^2

on the area of the ball with (x^2 + y^2 + z^2 <= 1) ....

can please someone help me. just wanna check if i did it right...

thank you guys!!!

ps: sorry if my expressions arn't correct... hope you understand what i mean tho.... im swiss

- Aug 24th 2007, 12:27 PM #2

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- Aug 24th 2007, 12:40 PM #3

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well, i looked first if it there is a max/min in the inside of the ball:

fx = 1

fy = 1

fz = 2z

it has to be: fx=fy=fz=0 but thats notpossible, so there is no max/min in the inside.

outside (surface):

transform x, y, z into spherical-coordinats:

x = sin(p)cos(q)

y = sin(p)sin(q)

z = cos(p)

than i wrote the function (f=x+y+z^2) new:

f(x(p,q), y(p,q), z(p,q) = .....

and than i did: df/dp and df/dq

df/dp=df/dq=0 for max/min, so i got for the value +-√2

and for the place maxx,y,z)=( (√2)/2 , (√2)/2 , 0)

min: ( -(√2)/2 , -(√2)/2 , 0)

is that right???? please say yes buddy

- Aug 24th 2007, 12:54 PM #4

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I did a different problem, I maximized

**on the boundary**.

But you wanted to maximize the boundary including the interior. The boundary can be done with Lagrange Multipliers like in my first post. The interior is found by doing . Which as you said is impossible. Hence look at the boundary.

- Aug 24th 2007, 12:58 PM #5

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- Aug 24th 2007, 01:01 PM #6

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- Aug 24th 2007, 01:02 PM #7

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- Aug 24th 2007, 01:06 PM #8

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- Aug 24th 2007, 02:20 PM #9

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- Aug 24th 2007, 02:50 PM #10

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