Sect. 14.7 Enhanced Assignment

Instructor: Shijun Zheng Course: Spring 2023 - Calculus III (MATH2243-A-B) Student: Jalen Wilson Date: 07/11/24 Assignment: Sect. 14.7 Enhanced Assign 1. Find all the local​maxima, local​minima, and saddle points of the function. 2 2 ​f(x,y) = − 7x − 8xy − 4y − 190x − 136y + 3 Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. A local maximum occurs at ( − 9, − 8) . (​ Type an ordered pair. Use a comma to separate answers as​needed.) The local maximum​value(s) is/are 1402 . ​(Type an exact answer. Use a comma to separate answers as​needed.) B. There are no local maxima. Select the correct choice below​and, if​necessary, fill in the answer boxes within your choice. A. A local minimum occurs at . ​(Type an ordered pair. Use a comma to separate answers as​needed.) The local minimum​value(s) is/are . ​(Type an exact answer. Use a comma to separate answers as​needed.) B. There are no local minima. Find the saddle points. Select the correct choice below​and, if​necessary, fill in the answer box within your choice. A. A saddle point occurs at . ​(Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no saddle points. 2. Find all the local​maxima, local​minima, and saddle points of the function. 2 ​f(x,y) = x + xy + 2x + 5y − 4 Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. A local maximum occurs at . ​(Type an ordered pair. Use a comma to separate answers as​needed.) The local maximum​value(s) is/are . ​(Type an exact answer. Use a comma to separate answers as​needed.) B. There are no local maxima. Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. A local minimum occurs at . ​(Type an ordered pair. Use a comma to separate answers as​needed.) The local minimum​value(s) is/are . ​(Type an exact answer. Use a comma to separate answers as​needed.) B. There are no local minima. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. A saddle point occurs at ( − 5,8) . ​(Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no saddle points. 3. Find all the local​maxima, local​minima, and saddle points of the function. f(x,y) = 6y sin (x) Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. A local maximum occurs at ​, where n is any integer. (​ Type an ordered pair. Type an expression using n as the variable. Use a comma to separate answers as​needed.) The local maximum​value(s) is/are . ​(Type an exact answer. Use a comma to separate answers as​needed.) B. There are no local maxima. Select the correct choice below​and, if​necessary, fill in the answer boxes to complete your choice. A. A local minimum occurs at ​, where n is any integer. (​ Type an ordered pair. Type an expression using n as the variable. Use a comma to separate answers as​needed.) The local minimum​value(s) is/are . ​(Type an exact answer. Use a comma to separate answers as​needed.) B. There are no local minima. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. A saddle point occurs at (nπ,0) ​, where n is any integer. (​ Type an ordered pair. Type an expression using n as the variable. Use a comma to separate answers as​needed.) B. There are no saddle points. 4. Solve by the substitution method. 8x + 7y = − 42 x = − 1 − 3y Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The solution set of the system is ( − 7,2) ​(Simplify your answer. Type an ordered pair.​) . B. There are infinitely many solutions. C. The solution set is the empty set. 5. Solve the equation by the​zero-factor property. 2 c − 9c + 18 = 0 The solution set is 3,6 . ​(Use a comma to separate answers as​needed.) 6. If fx (a,b) = fy (a,b) = ​0, must f have a local maximum or minimum at​(a,b)? Give reasons for your answer. Choose the best conclusion below. A. ​Yes, f must have a local minimum at​(a,b). Neither fx nor fy have negative signs in front of them in the​problem, so the First Derivative Test says they must be positive everywhere they are not​zero, and zero is smaller than all positive​numbers, so it is a minimum. B. N ​ o, it is impossible to tell what kind of point​(a,b) is. The First Derivative Test says to check the second partial​derivatives, but if fx and fy are both 0 at​(a,b), then all of fxx ​, fyy and fxy 2 must also be 0 at​(a,b), meaning fxx fyy − f xy = 0. Thus the Second Derivative Test fails to show what kind of point​(a,b) is. C. N ​ o, f could also have a saddle point at​(a,b). The First Derivative Test says that a point in the domain of a function​f(a,b) where both fx and fy are zero is a critical​point, but it does not specify what kind of critical point the point is. D. Y ​ es, f must have a local maximum at​(a,b). The First Derivative Test shows that all partial derivatives are​negative, and zero is larger than all negative​numbers, so it is a maximum. 7. 2 2 For the functions w = − 4x + 5y , x = cos t​, and y = sin t​, express dw dt in terms of t and differentiating directly with respect to t. Then evaluate Express dw dt = dt t= dt dw dt at t = as a function of t. 9 sin 2t Evaluate dw dw as a function of​t, both by using the chain rule and by expressing w dw dt π = 2 at t = π 2 0 . ​(Type an exact​answer, using radicals as​needed.) π 2 .