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pallavi pradeep bhardwaj Grade: 12
        

 Let g (x) = ∫0x f (t) dt, where f is such that 1/2 ≤ f(t) ≤ 1 for t [0,1] and


0 ≤ f(t) ≤1/2 for t [1,2]. Then g (2) satisfies the inequality:

7 years ago

Answers : (1)

Ramesh V
70 Points
										

g(2) = ∫02 f (t) dt = ∫01 f (t) dt   +  ∫12 f (t) dt


1/2 ≤ f(t) ≤ 1 for t ∈ [0,1] and 0 ≤ f(t) ≤1/2 for t ∈[1,2].


 011/2.dt ≤ ∫01 f (t).dt ≤ ∫01 1.dt   and 12 0.dt ≤ ∫12 f (t) dt ≤∫12 1/2.dt


1/2 ≤ ∫01 f (t).dt ≤  1   and     0 ≤ ∫12 f (t) dt ≤ 1/2


on adding above gives,


1/2 ≤ ∫01 f (t).dt + ∫12 f (t) dt ≤ 3/2


1/2 ≤ g(2) ≤ 3/2


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Regards,
Ramesh  
IIT Kgp - 05 batch







7 years ago
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