Question
Asked Oct 27, 2019
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Let f(x,y)= 3+cos(x) sin(2y)

a) Show that the function has a stationary point at (0,Pi/4).

b) Is (0, Pi/4) a local maximum, minimum or saddle point? Justify your answer mathematically.

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Expert Answer

Step 1

To analyze the critical point (0,pi/4) of the given function of two variables

Step 2

By definition, a stationary point is where both partial derivatives are 0.  a) is verified ,as shown. (suffixes denote partial derivatives).

f(x, y) 3+cos xsin 2y
f(x, y)-sinsin 2y
f,(0,)sin0sin
= 0
2
f,(x, y) 2cosxrcos 2y
(0, 2cos 0 cos= 0
2
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Image Transcriptionclose

f(x, y) 3+cos xsin 2y f(x, y)-sinsin 2y f,(0,)sin0sin = 0 2 f,(x, y) 2cosxrcos 2y (0, 2cos 0 cos= 0 2

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Step 3

b) to analyze further, we need the criter...

Let r fr,s ft= f
Criteria
1Drt-s20,r<0local max
2)rt-s20,r> 0=> local min
3)rt -s20,
saddle point
help_outline

Image Transcriptionclose

Let r fr,s ft= f Criteria 1Drt-s20,r<0local max 2)rt-s20,r> 0=> local min 3)rt -s20, saddle point

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Math

Calculus

Derivative