Application of DerivativeHard
Question
The function f(x) = sin4 x + cos4 x increases, if
Options
A.0 < x < 
B.
< x < 
< x < C. < x < 
< x < D. < x < 
< x < Solution
Given, f(x) = sin4 x + cos4 x
On differentiating w. r. t. x, we get
f′(x) = 4sin3 x cos - 4sin3 x sin x
= 4sin x cos x(sin2x - cos2x)
= 2sin 2x(- cos 2x)
= - sin 4x
Now, f′(x) > 0, if sin 4x < 0
⇒ π < 4x <2π ⇒
< x < 
.....(i)
⇒ Option (a) is not proper subset of Eq. (i), so it is not cprrect and (a) is wrong.
Now,
< x < 
Since, option (b) is the proper subset of Eq. (i). so it is correct.
On differentiating w. r. t. x, we get
f′(x) = 4sin3 x cos - 4sin3 x sin x
= 4sin x cos x(sin2x - cos2x)
= 2sin 2x(- cos 2x)
= - sin 4x
Now, f′(x) > 0, if sin 4x < 0
⇒ π < 4x <2π ⇒
< x < .....(i) ⇒ Option (a) is not proper subset of Eq. (i), so it is not cprrect and (a) is wrong.
Now,
< x < Since, option (b) is the proper subset of Eq. (i). so it is correct.
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