Transcription of Solutions to Exam 1 - Mathematics
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Solutions to Exam 1 - Mathematics
Solutions to Exam 11. Letf(x, y) =x2+y2+xy x+y.(a) Show thatx= 1, y= 1 is the only critical point :To find the critical point, setfx= 0 andfy= 0 to obtain theequations 2x+y 1 = 0,2y+x+ 1 = 0 which has the solutionx= 1, y= 1.(b) Use the second derivative test to show thatx= 1, y= 1 is a local mini-mum (and thus an absolute minimum it is the only critical point) :Sincefxx= 2 andD=fxxfyy f2xy= 2 2 1 = 3>0, thesecond derivative test says thatfis a local minimum.(c) Letz=L(x, y) be the equation of the tangent plane ofz=f(x, y) atthe critical point. Without evaluating any integrals, explain why the followinginequality holds: Rf(x, y)dA RL(x, y)dAwhereRis any rectangle [a, b] [c, d].Solution:The tangent plane atx= 1, y= 1 is parallel to thexy-plane(sincefx= 0 =fyat that point) and has the equation of the formL(x, y) =kwherekis a constant equal tof(1, 1).
equations 2x+y ¡1 = 0;2y +x+1 = 0 which has the solution x = 1;y = ¡1. (b) Use the second derivative test to show that x = 1 ;y = ¡ 1 is a local mini- mum (and thus an absolute minimum it …
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