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
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.
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
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.