Find the critical points of
:
![\dfrac{\partial f}{\partial x}=-2x=0\implies x=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D-2x%3D0%5Cimplies%20x%3D0)
![\dfrac{\partial f}{\partial y}=y-4y^3=y(1-4y^2)=0\implies y=0\text{ or }y=\pm\dfrac12](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3Dy-4y%5E3%3Dy%281-4y%5E2%29%3D0%5Cimplies%20y%3D0%5Ctext%7B%20or%20%7Dy%3D%5Cpm%5Cdfrac12)
All three points lie within
, and
takes on values of
![\begin{cases}f(0,0)=4\\f\left(0,-\frac12\right)=\frac{65}{16}\\f\left(0,\frac12\right)=\frac{65}{16}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Df%280%2C0%29%3D4%5C%5Cf%5Cleft%280%2C-%5Cfrac12%5Cright%29%3D%5Cfrac%7B65%7D%7B16%7D%5C%5Cf%5Cleft%280%2C%5Cfrac12%5Cright%29%3D%5Cfrac%7B65%7D%7B16%7D%5Cend%7Bcases%7D)
Now check for extrema on the boundary of
. Convert to polar coordinates:
![f(x,y)=f(\cos t,\sin t)=g(t)=4-\cos^2-\sin^4t+\dfrac12\sin^2t=3+\dfrac32\sin^2t-\sin^4t](https://tex.z-dn.net/?f=f%28x%2Cy%29%3Df%28%5Ccos%20t%2C%5Csin%20t%29%3Dg%28t%29%3D4-%5Ccos%5E2-%5Csin%5E4t%2B%5Cdfrac12%5Csin%5E2t%3D3%2B%5Cdfrac32%5Csin%5E2t-%5Csin%5E4t)
Find the critical points of
:
![\dfrac{\mathrm dg}{\mathrm dt}=3\sin t\cos t-4\sin^3t\cos t=\sin t\cos t(3-4\sin^2t)=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20dt%7D%3D3%5Csin%20t%5Ccos%20t-4%5Csin%5E3t%5Ccos%20t%3D%5Csin%20t%5Ccos%20t%283-4%5Csin%5E2t%29%3D0)
![\implies\sin t=0\text{ or }\cos t=0\text{ or }\sin t=\pm\dfrac{\sqrt3}2](https://tex.z-dn.net/?f=%5Cimplies%5Csin%20t%3D0%5Ctext%7B%20or%20%7D%5Ccos%20t%3D0%5Ctext%7B%20or%20%7D%5Csin%20t%3D%5Cpm%5Cdfrac%7B%5Csqrt3%7D2)
![\implies t=n\pi\text{ or }t=\dfrac{(2n+1)\pi}2\text{ or }\pm\dfrac\pi3+2n\pi](https://tex.z-dn.net/?f=%5Cimplies%20t%3Dn%5Cpi%5Ctext%7B%20or%20%7Dt%3D%5Cdfrac%7B%282n%2B1%29%5Cpi%7D2%5Ctext%7B%20or%20%7D%5Cpm%5Cdfrac%5Cpi3%2B2n%5Cpi)
where
is any integer. There are some redundant critical points, so we'll just consider
, which gives
![t=0\text{ or }t=\dfrac\pi3\text{ or }t=\dfrac\pi2\text{ or }t=\pi\text{ or }t=\dfrac{3\pi}2\text{ or }t=\dfrac{5\pi}3](https://tex.z-dn.net/?f=t%3D0%5Ctext%7B%20or%20%7Dt%3D%5Cdfrac%5Cpi3%5Ctext%7B%20or%20%7Dt%3D%5Cdfrac%5Cpi2%5Ctext%7B%20or%20%7Dt%3D%5Cpi%5Ctext%7B%20or%20%7Dt%3D%5Cdfrac%7B3%5Cpi%7D2%5Ctext%7B%20or%20%7Dt%3D%5Cdfrac%7B5%5Cpi%7D3)
which gives values of
![\begin{cases}g(0)=3\\g\left(\frac\pi3\right)=\frac{57}{16}\\g\left(\frac\pi2\right)=\frac72\\g(\pi)=3\\g\left(\frac{3\pi}2\right)=\frac72\\g\left(\frac{5\pi}3\right)=\frac{57}{16}\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dg%280%29%3D3%5C%5Cg%5Cleft%28%5Cfrac%5Cpi3%5Cright%29%3D%5Cfrac%7B57%7D%7B16%7D%5C%5Cg%5Cleft%28%5Cfrac%5Cpi2%5Cright%29%3D%5Cfrac72%5C%5Cg%28%5Cpi%29%3D3%5C%5Cg%5Cleft%28%5Cfrac%7B3%5Cpi%7D2%5Cright%29%3D%5Cfrac72%5C%5Cg%5Cleft%28%5Cfrac%7B5%5Cpi%7D3%5Cright%29%3D%5Cfrac%7B57%7D%7B16%7D%5Cend%7Bcases%7D)
So altogether,
has an absolute maximum of 65/16 at the points (0, -1/2) and (0, 1/2), and an absolute minimum of 3 at (-1, 0).
<span>f(x)= -2x +1 for f(3)
f(3) = -2(3) + 1 = -5
answer
</span><span>A)-5</span>
You answer is false because the 25 is almost equal to 28 have a nice day
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