Prove these examples are correct: a) What is the area of the largest rectangle that fits inside of the ellipse x^{2} + 2y^{2} = 1? b) Prove the following: Let c in (a, b). If f is continuous on [a, b], differentiable on (a, b)?

Elleanor Mckenzie 2021-02-25 Answered
Prove these examples are correct:
a) What is the area of the largest rectangle that fits inside of the ellipse
x2 + 2y2=1?
b) Prove the following: Let c in (a, b). If f is continuous on [a, b], differentiable on (a, b)?
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Velsenw
Answered 2021-02-26 Author has 91 answers

a) Сonsider this figure:
image
From the figure it can be seen that:
Area (A)=4xy
And also,
x2 + 2y2=1
x=1  2y2
We've taken the positive value since we chose this point to be in the first quadrant
So now deciding:
A=4xy
A=4y1  2y2
Differentiating the above function with respect to "y":
dAdy=ddy(4y1  2y2)
dAdy=4yddy1  4y2 + 1  4y2ddy(4y)
dAdy=4y121  4y2(8y) + 41  4y2
dAdy=4[8y2 + 2  8y221  4y2]
dAdy=4[2  16y221  4y2]
For maximize the area:
Put,
dAdy=0
dAdy=4[2  16y221  4y2]
[2  16y221  4y2]=0
2  16y2=0
y2=18
y=18
Corresponding to this,
x=1  2 × 18
x=1  14
x=34
x=32
Hence the maximum area:
Area (A)=4xy
Areamax=4 × 32 × 18
Areamax=2 × 322
Areamax=32
b)Prove the following: Let c in (a, b). If f is continuous on [a, b], differentiable on (a, b), and:
limx  x f(x)=L then f(c)=L
Properties used
limh  0 f(x + h)  f(x)h=f(x)
Proof is given below:
Since:
limx  cf(x)=L
By using the property:
limx  c [limh  0f(x + h)  f(x)h]=L

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