(1 pt) Let Find all critical points and classify them as local maxima, local min

Question

(1 pt) Let Find all critical points and classify them as local maxima, local minima, saddle points, or none of these. critical points: f(x,y) 1+2-cos(4y). (give your points as a comma separated list of (xy) 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+x2 - cos(4y)

for extrima point we have to calculate df/dx and df/dy and equate it to 0

df/dx = 2x = 0 .......(i)

df/dy = 4sin(4y) = 0 ......(ii)

critical points :-

from (i) x = 0

from (ii) y = n*pi/4 wher n = 1,2,3,4.......

local maxima at (x,y) = (0,-pi/4)(0,pi/4),(0,3pi/4),(0,5pi/4),(0,7pi/4)............

local minima at (x,y) = (0,-2pi/4),(0,2pi/4),(0,4pi/4),(0,6pi/4),(0,8pi/4).........

calculation of saddle point :-

for saddle point df/dx = 0 ,df/dy = 0 ........(iii) and

(d2f/dx2)*(d2f/dy2) - (d2f/dxdy)2 < 0 .........(a)

d2f/dx2 = 2

d2f/dy2= 16cos(4y)

d2f/dxdy = 0

by putting all these values in equation (a) we get 32cos(4y) - 0 < 0

cos(4y) < 0

y =(-pi/8 to -3pi/8),( pi/8 to 3pi/8), (5pi/8 to 7pi/8), (9pi/8 to 11pi/8).........(iv)

now from (iii) $ (iv) saddle points are (0,-pi/4),(0,pi/4),(0,3pi/4),(0,5pi/4)..........

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