if f(x)=sin 4 x + cos 4 x, then what is the range of f(x)

if f(x)=sin4x + cos4x, then what is the range of f(x)


4 Answers

Vijay Mukati
askIITians Faculty 2590 Points
7 years ago
Dear Student,

Differnetiate the functions and equate the result with zero to get the point of minima and maximua. You will get x = 0, 45 and 90 degree.

From here we can easily find the range and it will come out to be [1/2, 1]

Chirag Jain
36 Points
7 years ago
Sir, the answer is [3/4,1] as stated in the book and the book is Objective Maths by Purshottam Kumar Sharma
Chirag Jain
36 Points
7 years ago
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123 Points
7 years ago
in this question we have to find the range for \sin ^4x + \cos ^4x
now,\sin ^4x + \cos ^4x = (\sin ^2x)^2+(\cos ^2x)^2
\Rightarrow \sin ^4x + \cos ^4x = (\sin ^2x + \cos ^2x)^2-2\sin ^2x\cos ^2x
\Rightarrow \sin ^4x + \cos ^4x = 1-\frac{1}{2}(4\sin^2x\cos^2x)
\Rightarrow \sin ^4x + \cos ^4x = 1-\frac{1}{2}\sin^22x
as \sin^2\theta have a maximum value of 1 and a minimum value of 0
so, minimum value of \sin ^4x + \cos ^4x = 1-\frac{1}{2}(1) = \frac{1}{2}
and maximum value of \sin ^4x + \cos ^4x = 1-\frac{1}{2}(0)=1
so, range of f(x)=[1/2,1]
I think there is any mistake in the book. As you can see the detailed solution of the question and if there is any other doubt related to this question you can ask

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