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Question

Which of the following is true for α,β(0,) and αβ

A
2α+2β+13>2α+2β3
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B
lnα+2lnβ3ln(α+2β3)
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C
tan1α+2tan1β3>tan1(α+2β3)
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D
None of these
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Solution

The correct option is A 2α+2β+13>2α+2β3
Let f1(x)=2x,f2(x)=lnx,f3(x)=tan1x,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.

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