Calculus- Early Transcendentals, 2021a

13.7. Maxima and Minima 475
so there is a local minimum at (0,0), and there are no other possibilities.
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Example 13.27: Extrema on a Hyperbolic Paraboloid
Find all local maxima and minima for f (x,y)=x 2 − y 2 .
Solution. The derivatives:
f x = 2x f y = −2y f xx = 2 f yy = −2 f xy = 0.
Again there is a single critical point, at (0,0),and
D(0,0)= f xx (0,0) f yy (0,0) − f xy (0,0) 2 = 2 ·−2 − 0 = −4 < 0,
so there is neither a maximum nor minimum there, and so there are no local maxima or minima. The
surface is shown in Figure 13.9.
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7.5
5.0
2.5
0.0
−5.0
−2.5
−2.5
−5.0
0.0
y
5.0
2.5
0.0
x
−2.5
2.5
5.0
−5.0
Figure 13.9: A saddle point, neither a maximum nor a minimum.
Example 13.28: Finding Extrema
Find all local maxima and minima for f (x,y)=x 4 + y 4 .
Solution. The derivatives:
f x = 4x 3 f y = 4y 3 f xx = 12x 2 f yy = 12y 2 f xy = 0.
Again there is a single critical point, at (0,0),and
D(0,0)= f xx (0,0) f yy (0,0) − f xy (0,0) 2 = 0 · 0 − 0 = 0,