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What is max and min value of A

A= sin2x + cos4x

Suneil P. | Knowledgeable and Passionate University of Pennsylvania Math TutorKnowledgeable and Passionate University ...
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It may be worthwhile in general to confirm via the second derivative test as such:

Re-writing dA/dx as -sin(2x)cos(2x)= -sin(4x)/2, we can easily compute the second derivative: -2cos(4x)

At x=0, the second derivative is <0, and thus this is indeed a maximum-generating point
At x= pi/4, the second derivative is >0 and thus this is indeed a minimum-generating point

(no boundary value conditions stated)
PIYUSH L. | Maths tutoring for middle school to college maths studentsMaths tutoring for middle school to coll...
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A=sin2x + (1-sin2x)2
A= sin2x + 1 -2sin2x + sin4x
A = sin4x - sin2x + 1

Now to find the max and min vale of A we will find dA/dx and equal it to 0

dA / dx = 4sin3xcosx - 2sinxcosx = 0

2sinxcox (2sin2x - 1) = 0
Sin2x (2sin2x-1 ) = 0

So Either Sin2x=0 or 2sin2x-1 = 0

if sin2x=0, we have x=0,90,180 degree and so on
if 2sin2x-1 = 0, we have x=45, 135, 225 degree and so on

So  when x = 0 degree , A = 1
when x = 45 degree, A = 3/4

Max and min value of A is 1 and 3/4 respectively