Let f(x,y)=1+x^2cos(5y). Find all critical points and classify them as local max
ID: 2861049 • Letter: L
Question
Let
f(x,y)=1+x^2cos(5y).
Find all critical points and classify them as local maxima, local minima, saddle points, or none of these.
critical points: __________
(give your points as a comma separated list of (x,y) coordinates. if your answer includes points that occur at a sequence of values, e.g., at every odd integer, or at any constant multiple of another value, use m for any non-zero even integer, n for any non-zero odd integer, and/or k for other arbitrary constants.)
classifications: ___________
(give your answers in a comma separated list, specifying maximum, minimum, saddle point, or none for each, in the same order as you entered your critical points)
Please solve, and show steps.
Explanation / Answer
given
f(x,y)=1+x^2cos(5y).
For critical points fx =0 ,fy =0
fx =2x =0 =>x =0
fy =5sin(5y)=0 =>5y =mpi,npi
y =mpi/5 ,npi/5,0
critical points are (x,y)=(0,0),(0,mpi/5),(0,npi/5)
fxx =2
fyy=25cos(5y)
fxy =0
D=fxxfyy -fxy2
at (x,y)=(0,mpi/5)
fxx =2
fyy=25cos(0)=25
fxy =0
D=50 >0 minimum
at (x,y)=(0,mpi/5)
fxx =2
fyy=25cos(5mpi/5)=25
fxy =0
D=50 >0 minimum
at (x,y)=(0,npi/5)
fxx =2
fyy=25cos(5npi/5)=-25
fxy =0
D=-50 >0 saddle points
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