Minimizing a multivariable function given restraint I want to minimize the following function:
$$J(x, y, z) = x^a + y^b + z^c$$
I know I can easily determine the minimum value of $J$ using partial derivative. But I have also the following condition:
$$ x + y + z = D$$
How can I approach now?

This is an easy example of using Lagrange multiplier.
If you reformulate your constraint as $C(x,y,z) = x+y+z-D=0$, you can define
$L(x,y,z,\lambda) := J(x,y,z)-\lambda \cdot C(x,y,z)$
If you now take the condition $\nabla L=0$ as necessary for your minimum you will fulfill
$$\frac{\partial L}{\partial x}=0 \\ 
\frac{\partial L}{\partial y}=0 \\
\frac{\partial L}{\partial z}=0 \\
$$ 
Which are required for your minimum and
$$
\frac{\partial L}{\partial \lambda}=C(x,y,z)=0 \\
$$
as your constraint.
