Question

When doing langrage multipliers for calculus 3... lets say i already found my critical points and i got a value from it. how do i know if it is a min or max value?

Answer 1

One possible manner is the Hessian (2nd Derivative Test)

Explanation:

Typically to check if the critical points are mins or maxes, you will often use the Second Derivative Test, which requires you to find 4 partial derivatives, assuming `f(x,y)`:

`f_{"xx"}(x,y)`, `f_{"xy"}(x,y)`, `f_{"yx"}(x,y)`, and `f_{"yy"}(x,y)`

Note that if both `f_{"xy"}` and `f_{"yx"}` are continuous in a region of interest, they will be equal.

Once you have those 4 defined, you can then use a special matrix referred to as the Hessian to find the determinant of that matrix (which, confusingly enough, is often referred to as the Hessian as well), which will give you some information about the nature of the point. Thus, define the Hessian Matrix as:

`H = | (f_ {"xx"} color(white)(,aa) f_ {xy}) , (f_ {yx}color(white)(,aa) f_ {yy} )|`

Once you have that matrix established (and it will be a "function" matrix, since the contents will be functions of x and y), you can then take one of your critical points and evaluate the entire matrix determinant. Namely:

`det(H) = (f_ {"xx"}(x_0,y_0) * f_ {"yy"}(x_0,y_0)) - (f_ {"xy"}(x_0,y_0))^2`

Depending on the results of that calculation, you may learn the nature of the critical point:

If `H > 0`, there's a min/max at that point. Check the sign of `f_{"xx"}`. If it is positive, the point is a min. If it is negative, the point is a max. (This is analogous to the "traditional" 2nd derivative test for single-variable functions of x.)

If `H < 0`, there's a saddle point at that point.

If `H = 0`, the test is inconclusive and you must rely on other means, such as a graph of the function to visually determine.