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6.8. If f x; y 2x2 xy y2 , nd (a) @f =@x, and (b) @f =@y at x0 ; y0 directly from the de nition.

To avoid the hazards mentioned, force Visual Basic to require that you declare all variables before you use them. To do so, place the following line at the top of your code module: Option Explicit Alternatively, you can turn on explicit data type checking by choosing Tools Options and checking the Require Variable Declaration box, as shown in Figure 2-1.

The second-derivative test tells us nothing when f (c) = 0 and f (c) = 0. This is shown by the examples in Fig. 23-7, where, in each case, f (0) = f (0) = 0.

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2 x0 h 2 x0 h y0 y2 2x2 x0 y0 y2 0 0 0 h!0 h 4hx0 2h2 hy0 lim 4x0 2h y0 4x0 y0 lim h!0 h!0 h b @f f x0 ; y0 k f x0 ; y0 fy x0 ; y0 lim k!0 @y x0 ;y0 k lim 2x2 x0 y0 k y0 k 2 2x2 x0 y0 y2 0 0 0 k!0 k kx0 2ky0 k2 lim x0 2y0 k x0 2y0 lim k!0 k!0 k

Since the limits exist for all points x0 ; y0 , we can write fx x; y fx 4x y, fy x; y fy x 2y which are themselves functions of x and y. Note that formally fx x0 ; y0 is obtained from f x; y by di erentiating with respect to x, keeping y constant and then putting x x0 ; y y0 . Similarly, fy x0 ; y0 is obtained by di erentiating f with respect to y, keeping x constant. This procedure, while often lucrative in practice, need not always yield correct results (see Problem 6.9). It will work if the partial derivatives are continuous.

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Fig. 23-7 To distinguish among the four cases shown in Fig. 23-7, consider the sign of the derivative f just to the left and just to the right of the critical point. Recalling that the sign of the derivative is the sign of the slope of the tangent line, we have the four combinations shown in Fig. 23-8. These lead directly to Theorem 23-4. Theorem 23.4 (First-Derivative Test for Relative Extrema): Assume f (c) = 0. { , +} If f is negative to the left of c and positive to the right of c, then f has a relative minimum at c. {+, } If f is positive to the left of c and negative to the right of c, then f has a relative maximum at c. {+, +} and { , } If f has the same signs to the left and right of c, then f has an in ection point at c. 23.3 GRAPH SKETCHING

xy= x2 y2 x; y 6 0; 0 : Prove that (a) fx 0; 0 and fy 0; 0 both exist but 0 otherwise (b) f x; y is discontinuous at 0; 0 .

We are now equipped to sketch the graphs of a great variety of functions. The most important features of such graphs are: (i) Relative extrema (if any) (ii) In ection points (if any)

You can name variables almost any way you want, but some restrictions exist. Follow these rules in variable names: You must use a letter as the first character. You can t use a space; a period (.); an exclamation mark (!); or the characters @, &, $, or #

f h; 0 f 0; 0 0 lim 0 h!0 h h f 0; 0 f 0; 0 0 lim 0 fy 0; 0 lim k!0 k!0 k k

CHAP. 23]

mx2 m x!0 x!0 x m2 x2 1 m2 y!0 so that the limit depends on m and hence on the approach and therefore does not exist. Hence, f x; y

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