Application of Derivatives 1 Question 9

9. Consider f(x)=tan11+sinx1sinx,x0,π2.

A normal to y=f(x) at x=π6 also passes through the point

(a) (0,0)

(b) 0,2π3

(c) π6,0

(d) π4,0

(2016 Main)

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Answer:

Correct Answer: 9. H=φ,V=1,1

Solution:

  1. We have, f(x)=tan11+sinx1sinx,x0,π2

f(x)=tan1cosx2+sinx22cosx2sinx2=tan1cosx2+sinx2cosx2sinx2cosx2>sinx2 for 0<x2<π4=tan11+tanx21tanx2=tan1tanπ4+x2=π4+x2f(x)=12fπ6=12

Now, equation of normal at x=π6 is given by

yfπ6=2xπ6yπ3=2xπ6fπ6=π4+π12=4π12=π3

which passes through 0,2π3.



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