hw6-sol - Page 1 Assignment 6 Section 15.1 Problem 2(a...

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Unformatted text preview: Page 1 Assignment 6 Section 15.1 Problem 2 : (a) Point B. (b) Points E and G. (c) Points C, D and F. Problem 6 : at the origin, ℎ 0,0 ¡ = 1 . Since cos ? and cos ? are never above 1, the origin must be a local (and global) maximum. The second derivative test ¢ = ℎ ?? 0,0 ¡ℎ ?? 0,0 ¡ − £ℎ ?? 0,0 ¡¤ 2 = 1 > 0 And ℎ ?? 0,0 ¡ < 0 , so the origin is a local maximum. Problem 14 : at the critical point, ¥ ? = 8 ? − ? + ?¡ 3 = 0 ⇒ 8 ? = ? + ?¡ 3 ¥ ? = 8 ? − ? + ?¡ 3 = 0 ⇒ 8 ? = ? + ?¡ 3 Therefore, ? = ? = 0, ±1 . The critical points are 0,0 ¡ , 1,1 ¡ , − 1, − 1 ¡ . The discriminant is ¢ ? , ?¡ = − 64 + 48 ? + ?¡ 2 ¢ 0,0 ¡ = − 64 < 0 , so 0,0 ¡ is a saddle point. ¢ 1,1 ¡ = − 64 + 192 > 0 and ¥ ?? 1,1 ¡ = − 12 < 0 , so 1,1 ¡ is a local maximum. ¢ − 1, − 1 ¡ = − 64 + 192 > 0 and ¥ ?? − 1, − 1 ¡ = − 12 < 0 , − 1, − 1 ¡ is a local maximum. Problem 20 : we have ¥ ?? = 2 , ¥ ?? = 2, ¥ ?? = − 4 . The discriminant at the origin is ¢ 0,0 ¡ = 4 − 16 Thus, ¢ < 0 ⇒ < 4 , so the function has a saddle point at the origin when < 4 . ¦ ¢ > 0 ¥ ??...
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hw6-sol - Page 1 Assignment 6 Section 15.1 Problem 2(a...

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