Let f (x, y) = 1 + 3x^2 - cos (5y). Find all critical points and classify them a

Question

Let f (x, y) = 1 + 3x^2 - cos (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)

Explanation / Answer

f(x,y)=1+3x2-cos(5y)

fx=0

==>0+6x-0=0

==>x=0

fy=0

==>0+0+5sin(5y)=0

==>5y=0,npi,mpi , n for odd integer,m for even integer

==>y=mpi/5,npi/5

fxx=6

fyy=25cos(5y)

fxy=0

D=fxxfyy-fxy2

D=150cos(5y)

for (x,y)=(0,0) ,fxx=6,fyy=25cos(0)=25,D=150>0, fxx>0 ==> maximum at (x,y)=(0,0)

for (x,y)=(0,mpi/5) ,fxx=6,fyy=25cos(5mpi/5)=25,D=150>0, fxx>0 ==> maximum at (x,y)=(0,mpi/5)

for (x,y)=(0,npi/5) ,fxx=6,fyy=25cos(5npi/5)=-25,D=-150<0, fxx>0 ==>saddle point (x,y)=(0,npi/5)

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