The function `f(x) = sin ^(4)x+ cos ^(4)x ` increases, if (A) `0 lt x lt (pi)/(8)` (B) `(pi)/(4) lt x lt (3pi)/(8)` (C) `( 3pi)/(8) lt x lt (5pi)/(8)` (D) `( 5pi)/(8) lt x lt(3pi)/(4)`

Options

A. `0 lt x lt (pi)/(8)`
B. `(pi)/(4) lt x lt (3pi)/(8)`
C. `( 3pi)/(8) lt x lt (5pi)/(8)`
D. `( 5pi)/(8) lt x lt(3pi)/(4)`

Answer

B

Explanation

Given, `f (x) = sin ^(4)x + cos ^(4) x `
On differentiating w.r.t. x, we get
`f ' (x) = 4 sin ^(3) x cos x - 4 cos ^(3) x sinx `
`" " = 4 sin x cos x (sin ^(3)x - cos ^(2)x )`
` " " = 2 sin 2x (-cos 2 x)`
`" " = - sin 4x `
Now, `f ' (x) gt 0, if sin 4x lt 0`
`rArr " " pi lt 4x lt 2pi`
`rArr (pi)/(4) lt x lt (pi)/(2) " " ` ... (i)
`rArr ` Option (a) is not proper subset of Eq. (i), so it is not correct.
Now, ` " " (pi)/(4) lt x lt ( 3pi) /(8)`
Since, option (b) is the proper subset of Eq. (i), so it is correct.