Lagrange multipliers - finding maximum/minimum I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and its still unclear... I would appreciate responses in simple math terms/symbols.
The equation was $f(x, y, z) = xyz$, with constraint $x + y + z = 1$.
After few lines I obtain these C.P:
$(0, 0, 1)$, $(0, 1, 0)$, $(1, 0, 0)$ and $(1/3, 1/3, 1/3)$
When I saw online, the writer just said: 
$f(0, 0, 1) = 0$
$f(0, 1, 0) = 0$
$f(1, 0, 0) = 0 \leftarrow$ Minimums
$f(1/3, 1/3, 1/3) = 1/27 \leftarrow$ Maximum
I'm very confused because I've asked a tutor at my university and he told me to parametrize it and solve. But online it says I have to do Hessian for 3 variables to find whether it is maximum/minimum. (Which we haven't even looked at for 3 variables...!)
How do I know that it's a minimum/maximum? Is it because $1/27$ is bigger than $0$, so it is a maximum?

The method of Lagrange multiplier gives $(x,y,z)=(1/3,1/3,1/3)$ as the point of extremum. Since $(x,y,z)=(0,0,1)$ satisfies the constraint and $f(x,y,z)=0 < 1/27=f(1/3,1/3,1/3)$, it's safe to say that $1/27$ is the maximum.
I don't know how you got $(0,0,1),(0,1,0),(1,0,0)$ as critical points-$(1/3,1/3,1/3)$ should be the only point you get using Lagrange multiplier. Note that $f(0,0,1)$ isn't even the minimum since $f(-1,1,1)=-1<0$ and $-1+1+1=1$.
