Period of f(x)=sin⁴x + cos⁴x is ??? How???

Differentiate the function with respect to x.

d/dx of sin^4(x)+cox^4(x)=4sin^3(x)cos(x)-4cos^(x)sin(x)

Putting it equal to 0.

=4sin^3(x)cos(x)-4cos^(x)sin(x)=0

=4sin(x)cos(x)(sin^2(x)-cos^2(x))=0

sin(x)cos(x)=0 or sin^2(x)-cos^2(x)=0

If sin(x)=0,x=0 or cos(x)=0,x=(pie/2) (MAXIMA OF THE FUNCTION)

If sin^2(x)-cos^2(x)=0 then sin^2(x)=cos^2(x) or x=(pie)/4 or x=3(pie)/4 (MINIMA OF THE FUNCTION)

Since the function  has the same value after an interval of pie/2, So the period must be (pie)/2.

Hi Aditi,

As powers of sin and cos are even, and we have the sum of both sin and cos, they both can interchange and add.

ie A+B is same as B+A.

So period of the above function will be pi/2.

Best Regards,

Ashwin (IIT Madras).