Find the critical points of the following functions.Use the Second Derivative Te

Globokim8

Globokim8

Answered question

2021-10-13

Find the critical points of the following functions.Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum,or a saddle point. If the Second Derivative Test is inconclusive,determine the behavior of the function at the critical points.
f(x, y)=sin(2πx)cos(πy), for |x|12 and |y|12

Answer & Explanation

Asma Vang

Asma Vang

Skilled2021-10-14Added 93 answers

Step 1
We must determine the partial derivatives of the given function with respect to x and y in order to determine the critical point
fx(x,y)=2π(cos2πx)cos(πy)
fy(x,y)=π(sinπy)sin(2πx)
Step 2
Equating the derivatives to 0, to find the critical points
2π(cos2πx)cos(πy)=0
cos2πx=0
2πx=cos1(0)
2πx=π2
x=14
cosπy=0
πy=cos1(0)
πy=π2
y=12
Therefore the critical points from this is (14,a) and (b,12)
π(sinπy)sin(2πx)=0
sinπy=0
sinπy=0
πy=sin1(0)
y=0
sin(2πx)=0
2πx=sin1(0)
x=0
The critical points from this are (0,c) and (d,0)
Therefore the critical points of the given function are
(14,a), (b,12), (0,x) and (d,0)
Step 3
Let us use the second derivative test to determine whether the critical points correspond to a local maximum, a local minimum,or a saddle point.
D(x,y)=f×(x,y)fyy(x,y)(fxy(x,y))2
1. If D(x,y)>0 and f×(x,y)<0, then f has a local maximum value at (x,y)
2. If D(x,y)>0 and f×(x,y)>0, then f has a local minimum value at (x,y)
3. If D(x,y)<0 then f has a saddle point at (x,y)
4. If D(x,y)=0 then the test is inconclusive
Let us find f×(x,y), fyy(x,y), fxy(x,y)
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