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The following theorem may be useful in establishing maximums and minimums for the case of functions of two variables.

Maxima and Minima of two variable function in hindi

More investigation is necessary. Maxima and minima of functions subject to constraints. Let us set ourselves the following problem: Let F x, y and G x, y be functions defined over some region R of the x-y plane.

See Figure 1. The solution set i.

Let us now consider the same problem in three variables. Let F x, y, z and G x, y, z be functions defined over some region R of space. The problem then is to find the maximums of the function F x, y, z as evaluated on this spheroidal surface. Let us now consider another problem. Suppose instead of one side condition we have two. Let F x, y, z , G x, y, z and H x, y, z be functions defined over some region R of space.

Find the points at which the function F x, y, z has maximums subject to the side conditions.

## Morse : The critical points of functions and the calculus of variations in the large

Here we wish to find the maximum values of F x, y, z on that set of points that satisfy both equations 2 and 3. In Fig.

The intersection of the ellipsoid and the plane is the set F on which F x, y, z is to be evaluated. The above can be generalized to functions of n variables F x 1 , x 2 , Methods for finding maxima and minima of functions subject to constraints. Method of direct elimination. We can then substitute g x for y in F x, y and then find the maximums and minimums of F x, g x by standard methods. In some cases, it may be possible to do this kind of thing. As noted in Chapter 3, in multivariable calculus, the notion of differentiation manifests itself in several forms.

The simplest among these are the partial derivatives, which together constitute the gradient.

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