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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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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a) Сonsider this figure:From the figure it can be seen that:Area (A)=4xyAnd also,x2 + 2y2=1x=1 − 2y2We&#039;ve taken the positive value since we chose this point to be in the first quadrantSo now deciding:A=4xyA=4y1 − 2y2Differentiating the above function with respect to &quot;y&quot;:dAdy=ddy(4y1 − 2y2)dAdy=4yddy1 − 4y2 + 1 − 4y2ddy(4y)dAdy=4y121 − 4y2(−8y) + 41 − 4y2dAdy=4[−8y2 + 2 − 8y221 − 4y2]dAdy=4[2 − 16y221 − 4y2]For maximize the area:Put,dAdy=0dAdy=4[2 − 16y221 − 4y2][2 − 16y221 − 4y2]=02 − 16y2=0y2=18y=18Corresponding to this,x=1 − 2 × 18x=1 − 14x=34x=32Hence the maximum area:Area (A)=4xyAreamax=4 × 32 × 18Areamax=2 × 322Areamax=32b)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)=LProperties usedlimh → 0 f′(x + h) − f(x)h=f′(x)Proof is given below:Since:limx → cf′(x)=LBy using the property:limx → c [limh → 0f′(x + h) − f(x)h]=L

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)?

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