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If A = sin^2x + cos^4 x, then for all re...

If `A = sin^2x + cos^4 x`, then for all real x :

A

`(3)/(4) le A le (13)/(16)`

B

`(3)/(4) le A le 1`

C

`(13)/(16) le A le 1`

D

`1 le A le 2`

Text Solution

Verified by Experts

The correct Answer is:
B
`A = sin ^(2) x + cos ^(4) x `
`= 1- cos ^(2) x + cos ^(4)x`
`= 1 - cos ^(2) x (1- cos ^(2)x)`
` = 1- cos^(2)x sin ^(2) x `
` = 1- ( sin ^(2) 2x)/( 4)`
Now, `0 le sin ^(2) 2 x le 1`
`rArr - (1)/(4) le - ( sin^(2) 2x )/(4) le 0`
`rArr (3)/(4) le 1 - ( sin ^(2)2x)/(4x) le 1 `
`rArr 3//4 le A le 1`
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