Critical Points

In this section, we develop a method for finding the extrema-i.e.e, the maximum and minimum points-of a function of two variables. For reasons which will soon be apparent, this method is called the second derivative test.

To begin with, we say that a function f(x,y) has a local maximum at a point ( p,q) if there is a circle of radius R centered at ( p,q) such that

f(x,y)  £  f(p,q)

for all (x,y) in that circle. That is, f(p,q) is the maximum height of some small patch of the surface, although it may not be maximum overall.

Tan circle has radius R.

It follows that if  |h| < R, then f(p+h,q)  £  f(p,q) and thus

f(p+h,q) - f(p,q) £ 0

Dividing by h when 0< h < R and letting h approach 0 from the right yields

fx(p,q) =  lim  h®0+     f(p+h,q) - f(p,q) h   £   0

Conversely, dividing by h when -R< h < 0 and letting h approach 0 from the left yields

fx(p,q) =  lim  h®0-     f(p+h,q) - f(p,q) h   >   0

Consequently, it must follow that f_x(p,q) = 0. A similar argument shows that f_y(p,q) = 0. 

That is, the tangent plane to the graph of f(x,y) is horizontal at a local maximum or local minimum.

Similar results hold if f(x,y) has a local minimum at a point ( p,q) since this is equivalent to -f( x,y) having a local maximum at ( p,q) .      

Definition 8.1: The critical points of a function f(x,y) are those points ( p,q) for which

fx( p,q) = 0        and        fy(p,q) = 0

By the discussion above, the extrema of f( x,y) must occur at its critical points.       

EXAMPLE 1    Find the critical point(s) of

f( x,y) = x3-3xy+y3

Solution: The first partial derivatives are

fx( x,y) = 3x2-3y,        fy( x,y) = -3x+3y2

Setting f_x and f_y equal to zero leads to 2 simultaneous equations:

3x2-3y = 0,        -3x+3y2 = 0

Simplifying leads to y = x² and x = y², which implies that x = x⁴. Since x = x⁴ is the same as x⁴-x = 0, we obtain

x( x3-1)  =  0 x( x-1) ( x2+x+1)  =  0

which results in x = 0 and x = 1. Since y = x², we have x = 0 implies y = 0, while x = 1 implies y = 1. Thus, the critical points are (0,0) and ( 1,1) .

Graphic of Directional Derivative