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Question

Find the co-ordinates of the point (s) on the curve y=x21x2+1,x>0 such that tangent at these point (s)have the greatest slope.

A
(13,12).
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B
(13,12).
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C
(13,45).
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D
(3,12).
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Solution

The correct option is B (13,12).
y=x21x2+1=x2+12x2+1=12x2+1

y=12x2+1

S=slope =dydx=4x(x2+1)2

For maximum and minimum of S, dSdx=0

dSdx=4(x2+1)2.1x.2(x2+1).2x(x2+1)4

dSdx=4x2+14x2(x2+1)3

or dSdx=413x2(1+x2)3=0

x=13,13

Now consider change of sign at x=13,dSdx changes from +ve to -ve.
Hence S has maximum at x=13.
When x=13 the value of y=12
Hence the point is (13,12).

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