First find the critical points of <em>f</em> :
![f(x,y)=2x^2+3y^2-4x-5=2(x-1)^2+3y^2-7](https://tex.z-dn.net/?f=f%28x%2Cy%29%3D2x%5E2%2B3y%5E2-4x-5%3D2%28x-1%29%5E2%2B3y%5E2-7)
![\dfrac{\partial f}{\partial x}=2(x-1)=0\implies x=1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20x%7D%3D2%28x-1%29%3D0%5Cimplies%20x%3D1)
![\dfrac{\partial f}{\partial y}=6y=0\implies y=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D6y%3D0%5Cimplies%20y%3D0)
so the point (1, 0) is the only critical point, at which we have
![f(1,0)=-7](https://tex.z-dn.net/?f=f%281%2C0%29%3D-7)
Next check for critical points along the boundary, which can be found by converting to polar coordinates:
![f(x,y)=f(10\cos t,10\sin t)=g(t)=295-40\cos t-100\cos^2t](https://tex.z-dn.net/?f=f%28x%2Cy%29%3Df%2810%5Ccos%20t%2C10%5Csin%20t%29%3Dg%28t%29%3D295-40%5Ccos%20t-100%5Ccos%5E2t)
Find the critical points of <em>g</em> :
![\dfrac{\mathrm dg}{\mathrm dt}=40\sin t+200\sin t\cos t=40\sin t(1+5\cos t)=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dg%7D%7B%5Cmathrm%20dt%7D%3D40%5Csin%20t%2B200%5Csin%20t%5Ccos%20t%3D40%5Csin%20t%281%2B5%5Ccos%20t%29%3D0)
![\implies\sin t=0\text{ OR }1+5\cos t=0](https://tex.z-dn.net/?f=%5Cimplies%5Csin%20t%3D0%5Ctext%7B%20OR%20%7D1%2B5%5Ccos%20t%3D0)
![\implies t=n\pi\text{ OR }t=\cos^{-1}\left(-\dfrac15\right)+2n\pi\text{ OR }t=-\cos^{-1}\left(-\dfrac15\right)+2n\pi](https://tex.z-dn.net/?f=%5Cimplies%20t%3Dn%5Cpi%5Ctext%7B%20OR%20%7Dt%3D%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%2B2n%5Cpi%5Ctext%7B%20OR%20%7Dt%3D-%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%2B2n%5Cpi)
where <em>n</em> is any integer. We get 4 critical points in the interval [0, 2π) at
![t=0\implies f(10,0)=155](https://tex.z-dn.net/?f=t%3D0%5Cimplies%20f%2810%2C0%29%3D155)
![t=\cos^{-1}\left(-\dfrac15\right)\implies f(-2,4\sqrt6)=299](https://tex.z-dn.net/?f=t%3D%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%5Cimplies%20f%28-2%2C4%5Csqrt6%29%3D299)
![t=\pi\implies f(-10,0)=235](https://tex.z-dn.net/?f=t%3D%5Cpi%5Cimplies%20f%28-10%2C0%29%3D235)
![t=2\pi-\cos^{-1}\left(-\dfrac15\right)\implies f(-2,-4\sqrt6)=299](https://tex.z-dn.net/?f=t%3D2%5Cpi-%5Ccos%5E%7B-1%7D%5Cleft%28-%5Cdfrac15%5Cright%29%5Cimplies%20f%28-2%2C-4%5Csqrt6%29%3D299)
So <em>f</em> has a minimum of -7 and a maximum of 299.
Answer:
x = -24
Step-by-step explanation:
We gotta get rid of that 3 under x, so we multiply the reciprocal of 1/3, which is 3, if we multiply that on one side, we must also multiply on the other side.
3 · x/3 = -8 · 3
x = -24
Hope this helps! good luck on whatever you're doing :]
6x^2 is the correct answer to the problem.
Answer:
-11
Step-by-step explanation:
-9 +(-1) 2
-9 - 1 x 2
-9 - 2
-11
Answer:
44 cups = 27.5 gallons
Step-by-step explanation:
Using the conversion:
We can find what is 44 cups in gallons.
=> 1 cup = 0.625 gallons
=> 44 x 1 = 0.625 x 44
=> 44 cups = 0.625 x 44 gallons
=> 44 cups = 27.5 gallons
Therefore, 44 cups = 27.5 gallons.
Hoped this helped.