Question 1209923
Let f(x,y,z)=3x+4y+5z+x 
3
 + 
z
4x 
2
 y
​
 + 
xy 
2
 
z 
5
 
​
  and g(x,y,z)=x 
2
 +y 
2
 +z 
2
 −1=0.

We use the method of Lagrange multipliers to find the maximum value of f(x,y,z) subject to the constraint g(x,y,z)=0.

We need to solve the system of equations:

∇f=λ∇g
g(x,y,z)=0
First, let's find the gradients:

∇f=(3+3x 
2
 + 
z
8xy
​
 − 
x 
2
 y 
2
 
z 
5
 
​
 ,4+ 
z
4x 
2
 
​
 − 
xy 
3
 
2z 
5
 
​
 ,5− 
z 
2
 
4x 
2
 y
​
 + 
xy 
2
 
5z 
4
 
​
 )

∇g=(2x,2y,2z)

Now, we set ∇f=λ∇g:

3+3x 
2
 + 
z
8xy
​
 − 
x 
2
 y 
2
 
z 
5
 
​
 =2λx
4+ 
z
4x 
2
 
​
 − 
xy 
3
 
2z 
5
 
​
 =2λy
5− 
z 
2
 
4x 
2
 y
​
 + 
xy 
2
 
5z 
4
 
​
 =2λz
x 
2
 +y 
2
 +z 
2
 =1
This system of equations is highly nonlinear and difficult to solve analytically. However, we can make some observations.

If we consider the case where x,y,z>0, we have:

When x=0.6, y=0.8, and z is a small positive number, the term  
z
4x 
2
 y
​
  becomes very large.
When x,y are small positive numbers and z is close to 1, the term  
xy 
2
 
z 
5
 
​
  becomes very large.
This suggests that the maximum value might occur when x,y,z are all positive and have a certain relationship.

That's the only answer I can give for now!