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using the chain rule find the min and max points and their values of the composite function defined by z=x^2+y^2, x=sin(2t), y=cos(t)

This is what Ive got so far...
dz/dt= 2x*dx/dt+2y*dy/dt
=4sin2tcos2t-2sintcost

I understand the steps of computing the max and min I just can't seem to solve this problem. Its on a test I have due tomorrow so please help!!

You need to find dz/dt=0, so set
4sin2tcos2t-2sintcost =0
Expand using double-angle formulae:
8cos(t)sin(t)(cos(t)^2-sin(t)^2)-2cos(t)sin(t) = 0
Factorize:
-2cos(t)sin(t)(4sin(t)^2-4cos(t)^2+1)
Which gives
t=0, t=π/2, t=π and t=3π/2
when sin(t)=0 or cos(t)=0.
This leaves us with:
4sin(t)^2-4cos(t)^2+1=0
which can be reduced by the substitution
cos(t)^2=1-sin(t)^2
to a quadratic equation equivalent to
sin(t)=sqrt(3/8)

Now that leaves you to check each point as a maximum/minimum or inflection point using second derivatives.

vai toma no cu

vai te fude caralho nao fode nao vai estudar nao tem essa de ficar pedindo ajuda pr aprova na net ta xapada de nescau caraio

piranha fodida

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