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Calculus/Maxima minima using partial differentiation

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Expert: Scotto - 7/29/2008


Question
For a function such as y = ax + bz that depend on 2 or more variables (2 in this case), why is "Equating the partial differential of the function above with respect to each of independent variables (x and z in this case) and subsequently solving the thus generated simultaneous equations for the values of x and z" a sufficent condition to determine the maxima of the function.

Additionally, what is the corresponding sufficient condition to determine the minima of the same function?


Best regards


Deepak Agarwal

Answer
The reason that this provides a solution is that the partial derivative in each variable is like if the other variable were a constant.  To find the maximum with respect to one variable, you would take the derivative and set it equal to 0 as if the other variable were a constant.  This is what the  partial derivatives do with respect to x and z.

What is ended up with is two equations with two unknowns that are both equal to zero.

What this will tell  you is where the maximum, minimum, and any inflection points are at.

The second partial for each variable will say whether it is a maximum, minimum, or inflection point in that direction.

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