If a continuous function e−xf(x) has only one stationary point and attains its maximum in [1,3] at x=2, then which of the following(s) is/are correct.
A
f(x)<f′(x) for 1<x<2
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B
f(x)<f′(x) for 2<x<3
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C
f(x)>f′(x) for 1<x<2
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D
f(x)>f′(x) for 2<x<3
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Solution
The correct option is Df(x)>f′(x) for 2<x<3 For a continuous function in [1,3] and has a maxima at x=2
Let g(x)=e−xf(x) ⇒g′(x)>0 for 1<x<2 ⇒e−x(f′(x)−f(x))>0 for 1<x<2 ⇒f′(x)>f(x) for 1<x<2
⇒g′(x)<0 for 2<x<3 ⇒e−x(f′(x)−f(x))<0 for 2<x<3 ⇒f′(x)<f(x) for 2<x<3