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:
The horsepower required is 235440 watt.
Step-by-step explanation:
To find : What horsepower is required to lift an 8,000 pound aircraft six feet in two minutes?
Solution :
The horsepower formula is given by,
![W=\frac{mgh}{t}](https://tex.z-dn.net/?f=W%3D%5Cfrac%7Bmgh%7D%7Bt%7D)
Where, W is the horsepower
m is the mass m=8000 pound
g is the gravitational constant g=9.81
t is the time t= 2 minutes
h is the height h=6 feet
Substitute all values in the formula,
![W=\frac{8000\times 9.81\times 6}{2}](https://tex.z-dn.net/?f=W%3D%5Cfrac%7B8000%5Ctimes%209.81%5Ctimes%206%7D%7B2%7D)
![W=\frac{470880}{2}](https://tex.z-dn.net/?f=W%3D%5Cfrac%7B470880%7D%7B2%7D)
![W=235440](https://tex.z-dn.net/?f=W%3D235440)
Therefore, The horsepower required is 235440 watt.
<h2>I think it's <u>false</u></h2>
<h2><u><em>I</em><em> </em><em>hope</em><em> </em><em>it's</em><em> </em><em>helpfull </em><em>for</em><em> you</em></u></h2>