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)

Question

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!!

Answer 1

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.

Answer 2

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Answer 3

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Answer 4

piranha fodida