Which of the following is true for α,β∈(0,∞) and α≠β
A
2α+2β+13>2α+2β3
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B
lnα+2lnβ3≥ln(α+2β3)
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C
tan−1α+2tan−1β3>tan−1(α+2β3)
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D
None of these
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Solution
The correct option is A2α+2β+13>2α+2β3 Let f1(x)=2x,f2(x)=lnx,f3(x)=tan−1x,x>0
For concavity : As x>0 f′′1(x)=2x(ln2)2>0(i.e. upward concave),f′′2(x)=−1x2<0(i.e. downward concave),f′′3(x)=−2x1+x2<0(i.e. downward concave)
Now considering both cases of concavity :
Let the two points on the curve are A(α,f(α)) and B(β,f(β))
If M intersecting AB internally in ratio 2:1 ⇒M≡(α+2β3,f(α)+2f(β)3)
and N≡(α+2β3,f(α+2β3))
Case (a): Upward concave
Point N lies below M. ⇒f(α+2β3)<f(α)+2f(β)3
and
Case (b): Downward concave
Point N lies above M. ⇒f(α+2β3)>f(α)+2f(β)3
So, option (a) is correct.