Solução_Calculo_Stewart_6e

F.
202 ¤ CHAPTER 15 PARTIAL DERIVATIVES ET CHAPTER 14
TX.10
positive as well as points where f is negative, so the graph crosses its tangent plane (z =0)there and (π, π) isasaddlepoint.
D π
, π
3 3 =
9
> 0 and f π
4 xx , π
3 3 < 0 so f π , π
3 3 =
3 √ 3
is a local maximum while D 5π
, 5π
2 3 3 =
9
> 0 and
4
f 5π xx , 5π
3 3 > 0,sof 5π , 5π
3 3 = −
3 √ 3
is a local minimum.
2
25. f(x, y) =x 4 − 5x 2 + y 2 +3x +2 ⇒ f x(x, y) =4x 3 − 10x +3and f y(x, y) =2y. f y =0 ⇒ y =0, and the graph
of f x shows that the roots of f x =0are approximately x = −1.714, 0.312 and 1.402. (Alternatively, we could have used a
calculator or a CAS to find these roots.) So to three decimal places, the critical points are (−1.714, 0), (1.402, 0), and
(0.312, 0). Nowsincef xx =12x 2 − 10, f xy =0, f yy =2,andD =24x 2 − 20, wehaveD(−1.714, 0) > 0,
f xx (−1.714, 0) > 0, D(1.402, 0) > 0, f xx (1.402, 0) > 0,andD(0.312, 0) < 0. Therefore f(−1.714, 0) ≈−9.200 and
f(1.402, 0) ≈ 0.242 are local minima, and (0.312, 0) is a saddle point. The lowest point on the graph is approximately
(−1.714, 0, −9.200).
27. f(x, y) =2x +4x 2 − y 2 +2xy 2 − x 4 − y 4 ⇒ f x(x, y) =2+8x +2y 2 − 4x 3 , f y(x, y) =−2y +4xy − 4y 3 .
Now f y =0 ⇔ 2y(2y 2 − 2x +1)=0 ⇔ y =0or y 2 = x − 1 .Thefirst of these implies that f 2 x = −4x 3 +8x +2,
and the second implies that f x =2+8x +2
x − 1 2 − 4x 3 = −4x 3 +10x +1. From the graphs, we see that the first
possibility for f x has roots at approximately −1.267, −0.259,and1.526, and the second has a root at approximately 1.629
(the negative roots do not give critical points, since y 2 = x − 1 must be positive). So to three decimal places, f has critical
2
points at (−1.267, 0), (−0.259, 0), (1.526, 0), and(1.629, ±1.063). Nowsincef xx =8− 12x 2 , f xy =4y,
f yy =4x − 12y 2 ,andD =(8− 12x 2 )(4x − 12y 2 ) − 16y 2 ,wehaveD(−1.267, 0) > 0, f xx (−1.267, 0) > 0,
D(−0.259, 0) < 0, D(1.526, 0) < 0, D(1.629, ±1.063) > 0,andf xx (1.629, ±1.063) < 0. Therefore, to three decimal
places, f(−1.267, 0) ≈ 1.310 and f(1.629, ±1.063) ≈ 8.105 are local maxima, and (−0.259, 0) and (1.526, 0) are saddle
points. The highest points on the graph are approximately (1.629, ±1.063, 8.105).