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Pavel [41]
3 years ago
10

Help me plz thank you but try to put the right answer

Mathematics
1 answer:
Sedbober [7]3 years ago
3 0

Answer:

Option D is correct.

Step-by-step explanation:

Have a nice day. ;-)

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Find the absolute extrema for f(x,y)=4-x^2-y^4+1/2y^2 over the closed disk D:x^2+y^2 is less than or equal to 1
algol [13]

Find the critical points of f(x,y):

\dfrac{\partial f}{\partial x}=-2x=0\implies x=0

\dfrac{\partial f}{\partial y}=y-4y^3=y(1-4y^2)=0\implies y=0\text{ or }y=\pm\dfrac12

All three points lie within D, and f 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}

Now check for extrema on the boundary of D. 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

Find the critical points of g(t):

\dfrac{\mathrm dg}{\mathrm dt}=3\sin t\cos t-4\sin^3t\cos t=\sin t\cos t(3-4\sin^2t)=0

\implies\sin t=0\text{ or }\cos t=0\text{ or }\sin t=\pm\dfrac{\sqrt3}2

\implies t=n\pi\text{ or }t=\dfrac{(2n+1)\pi}2\text{ or }\pm\dfrac\pi3+2n\pi

where n is any integer. There are some redundant critical points, so we'll just consider 0\le t< 2\pi, 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

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}

So altogether, f(x,y) 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).

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<span>So the question is what are three equivalent ratios for 4/3, 12/14 and 6/9. The simplest way to get the equivalent ratio of some other ratio is to either multiply the nominator and the denominator by the same number or to divide the nominator and the denominator with the same number. I need to point out that it's not always possible to divide them and get a whole number. First: (4/3)*(2/2)=8/6, (4/3)*(3/3)=12/9 and (4/3)*(4/4)=16/12. Second: (12/14)/(2/2)=6/7, (12/14)*(2/2)=24/28 and (12/14)*(3/3)=36/42. Third: (6/9)/(3/3)=2/3, (6/9)*(2/2)=12/18 and finally (6/9)*(3/3)=18/27.</span>
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