Differential Equations 1 Question 22

22. Which of the following is true?

(a) g is increasing on (1,)

(b) g is decreasing on (1,)

(c) g is increasing on (1,2) and decreasing on (2,)

(d) g is decreasing on (1,2) and increasing on (2,)

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Solution:

  1. Here, f(x)=(1x)2sin2x+x20,x

and g(x)=1x2(t1)t+1logtf(t)dt

g(x)=2(x1)(x+1)logxf(x)+ve 

For g(x) to be increasing or decreasing.

Let φ(x)=2(x1)x+1logx

φ(x)=4(x+1)21x=(x1)x(x+1)2

φ(x)<0,x>1

φ(x)<φ(1)φ(x)<0

From Eqs. (i) and (ii), g(x)<0,x(1,)

g(x) is decreasing on x(1,).



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