Question 1
Find the absolute maximum and minimum values of f(x, y) = xy
3 e
-(x
2 + y
2) on the semi-infinite strip
![](data:image/png;base64, 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)
, if these extreme values exist.
◦ maximum
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAAqADIDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9U6KKKAOI+NfjHV/h58KvFXijRbWyu7zRdLu9R8u/dxHiGCSUDagyxLIq43LwxOeMHqtFvH1HR7G7kCrJPBHKwToCygnHtzXL/GfwNqnxN+F3iXwlpGsWegz65YzadLf3unverHDKjRybYlmhO/ax2kvgHBKsOK2/Bul6ponhbTNP1q/s9U1O2hEU15YWb2kMuOAVieWUpxjIMjc5PGcCly8r73Kk1aKW+t/wt+ptUUUVJIUUUUAeVf8ADJ3wQ/6I38P/APwl7H/41Xmnj74K/DrTfH+k+EPBP7Onwn8T6lNplxquoy6tZ22nQ6dEssMVurmLT7k7rhnuDGGCBhZz4LbGA9g1/wCLP/CO+O9N0a60DUItBub2PSZvElwvk2sN/NEZbeIbwN8TbVhMwO37Rc2sCebI8wt+HXTvDXjH4k/GvV/F9ra3Hh3Tl0fwldaZfR+fBdi2tm1FZHTGJPMbXBCICrZNuCCxkCoa9BpXdh/gr9lv4dX+lSyeL/gF8J9D1ITFY7fRNNttShaLauHMklhbkNuLjbsIAAO45IF7xD+y18IrHSZpdH+BPw71bUuFgtp9AsbaIsTjMknkMVQdSVR2x0VjxV74V+Gl+Cvwn16eLTJ7HSoZ77WLDw1GQf7NtTl0tIgCQuQpfYvyo0rIuVUE8L8LPif8XPjJ4SvtQu/AOgTeH9X0+G800axc28FleQygiS0aW1u78yjYwPntDEDgqYRu+WJPeK3SV/K/Tr2e19vS5tq9rtLzs7X/ABT+euztkaJ8NfhHN8OfFWva3+zn8PNM1nQL6bTG0200ixuba9uF2CMW9ybWMsjvKse5o1KsHBXK1qeHvhj4G8JfGHw9p2ofs7fD7w1HfSS3Gg+ItFs7Oe6gubYCYedGLSM27YUsjxySgFACVJFangT4GSeCL7T9LttI0/w9pOra4viS/wDD/h1G/sfTPstvGscEDeXGpZ7oQzFvLiL+W2IxtZq6bwlp3xPPxKuNY8T+GvCTWEhltbe/sfE91NPZWRIZUjtW05FLuyRmQmYbio5wiKNUrtX002+Xf12/XYLe63/VrvT7up65RRRUiKmraTY6/pV7pmp2VvqOm3sL211Z3cSywzxOpV43RgQyspIKkEEEg15f8Pf2bvDnhr4V/wDCEeL/AC/ilBJqE9/d6l4xtI7+61FzK32d7tpAwmmhtxBAJT1WBNoRQqL63RQO9tjnfBPw58J/DXTprDwh4X0bwrYzy+dLa6Jp8VnFJJgDeyxqoLYAGTzgCo/CXww8HeALrULrwv4T0Pw3c6g/mXk2kabDavctkndIY1Bc5JOWz1NdNRTu273F5BRRRSAKKKKAP//Z)
, minimum -
![](data:image/png;base64, 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)
◦ maximum
![](data:image/png;base64, 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)
, minimum -
![](data:image/png;base64, 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)
◦ maximum
![](data:image/png;base64, /9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAMCAgMCAgMDAwMEAwMEBQgFBQQEBQoHBwYIDAoMDAsKCwsNDhIQDQ4RDgsLEBYQERMUFRUVDA8XGBYUGBIUFRT/2wBDAQMEBAUEBQkFBQkUDQsNFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBQUFBT/wAARCAAsAB8DASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD9E/if8YIfhXbve3vhfxDq2jWyLLqOraXBC0Gnxk43yCSVJJAOSRAkrKBkgZGfQK8O/aG8I618T7eXwtaeFvEvmNC0mleJ9J8SLp+n2d1gGN7yFLlJZQkiIwXyLheAcDJxZ8QjxT8HJrj4h+IfFlx4m8OWVpc/8JJZR2620NnYpseG9toMtlrZEnadQ3mXAuZWUn7PaWgUXdar+v8AgenVavoPRr+v6v8Ap068r4qsB8QPid8RdTvLfx7rMPhy5sPC9h4a8G+J7zSvMmWzTUZ7xmjvLWANImpQw4mbI+xKVdjIEX1n4J6rout/DLR7vw8utppbGdFi8SXlxd6hFIs8iSxzS3EssjMsiuvMjABQFO0LXlngKe/1L4HTaxb+FPEHjK08ba1quttdeGtYj03VLjT7i+lk0u48+S5tnA+wCyQKZkkjjSOIqAhRfSvgR4E1P4bfDm10HUrqadYLq5eyt7i5a5lsrR5naC2edvmlaOMqpck85G5gAxSerVv6/p+ez1XUfR/16/13+7M/4U34u/6Lt8QP/AHw9/8AKqj/AIU34u/6Lt8QP/AHw9/8qq77xPrEukWCLaIsuo3cgtrON1JQykE7mxztVVZ29lOOcV87aX+2f4c+HnwY+G+rfEDVY7zxR4g0KPVZ4Y7uwsXaPCiWYC5ngRgGbAjjLSNztRsHGii+Vz6Jpfem/wAEm32KUW1df1/X5lq30uy8G+JdG+Htv8efiDok2E07TLaLw1o0GmApHlLSG5Gii2DqiYEKvuAXAXjFe1+BfBur+Evt39q+O/EHjX7Rs8v+3YNOi+zbd2fL+x2lvnduGd+77i42/NnzDXPix4f+IPj7QdKuNI8VtoGnXtrqFtdp4M1eW21G6Khrd1uUtTEkEZdXLs4y6DO1Fbf71RytRvL+tvx8u1u+ktq/u7WX6/pb8Ti/ix8IfDPxm8LvoviTTLG9RTvtbq70yzv3s5MY8yJLuGaIPjIy0bcE15r4F/Y60L4YaL4ZsvCHjPxR4au9Cgms49R06PTEku7aRxIYbiL7EYJcOu4SGLzcsx8w7jn36ioSs7r+v6WnppsNu6s9v6/ryeu4yJDHEiM7SsoALvjLH1OABn6AU+iimI//2Q==)
, minimum -
![](data:image/png;base64, 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)
◦ maximum
![](data:image/png;base64, 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)
, minimum 0
◦ no extreme values exist
Question 2
A function f(x, y, z) has continuous first partial derivatives at all points in 3-space. Its restriction to a closed, bounded domain R is identically zero on the boundary of R, but has the values 5 and -3 at two interior points of R. What is the smallest number of critical points that f must have in the interior of R?
◦ 0
◦ 2
◦ 4
◦ 3
◦ 1