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If sinx+xge|k|x^(2), AA x in [0,(pi)/(2)...

If `sinx+xge|k|x^(2), AA x in [0,(pi)/(2)]`, then the greatest value of k is

A

`(-2(2+pi))/(pi^(2))`

B

`(2(2+pi))/(pi^(2))`

C

can't be determined finitely

D

zero

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)=sinx+x,`
`rArr" "f'(x)=cos x+1 gt 0`
`rArr" "f(x)` is increasing function
Also `f''(x)=-sinx lt0`
`therefore" f is concave downward for " x in [0,(pi)/(2)]`
Now `g(x)=|k|x^(2)` is concave upward and increasing
So, if `g((pi)/(2))le f((pi)/(2))`, then `f(x)geg(x)`
`rArr" "1+(pi)/(2)ge|k|(pi^(2))/(4)`
`rArr" "k=[(-(2pi+4))/(pi^(2)),((2pi+4))/(pi^(2))]`
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