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Question

Find the point on the curve y=x1+x2, where the tangent to the curve has the greatest slope.

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Solution

y=x1+x2

dydx=(1+x2)2x2(1+x2)2

= 11+x22x2(1+x2)2=(1x)2(1+x2)2

d2ydx2=(1+x2)2[2x4x][(1+x)22x2][2(1+x2.2x](1+x2)4

=(1+x2)2[2x]+[2x2(1+x2)][4x(1+x2)](1+x2)4<0

dydx=0(1x2)=0x=1

y=11+1=12

Therefore, point is (x,y)=(1,12)


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