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Given A = {x : (pi)/(6) le x le (pi)/( 3...

Given `A = {x : (pi)/(6) le x le (pi)/( 3)} and f(x) = cos x - x ( 1+ x )`. Find ` f (A)`.

Text Solution

Verified by Experts

Given, `A = { x : (pi)/(6) le x le (pi)/(3)}`
and ` f (x) = cos x - x - x ^(2)`
` rArr f ' (x) = - sin x - 1 - 2x = - (sin x + 1 + 2 x )`
which is negative for ` x in [ ( pi)/(6), (pi)/(3)]`
` therefore " " f ' (x) lt 0`
or ` f (x)` is decreasing.
Hence, ` f (A) = [ f((pi)/(3)), f ((pi)/(6))]`
` " " = [ (1)/(2) - (pi)/(3) ( 1+ (pi)/(3)) , (sqrt 3 ) /( 2)- (pi)/(6) ( 1 + (pi)/(6)]`
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