`f(x) = sin^(4)x+cos^(4)x` in `[0,(pi)/(2)]`

Verify Rolle's theorem for each of the following functions: (i) f(x) = sin 2x " in " [0, (pi)/(2)] (ii) f(x) = (sin x + cos x) " in " [0, (pi)/(2)] (iii) f(x) = cos 2 (x - (pi)/(4)) " in " [0, (pi)/(2)] (iv) f(x) = (sin x - sin 2x) " in " [0, pi]

Find the points of local maxima and local minima, if any, of the following functions. Find also the local maximum and local minimum values : f(x)=sin^(4)x+cos^(4)x,0ltxlt(pi)/(2).

The minimum value of f(x)-sin^(4)x+cos^(4)x,0lexle(pi)/(2) is

int(sqrt(sin^(4)x+cos^(4)x))/(sin^(3)x cos x)dx,x in(0,(pi)/(2))

int(sqrt(sin^(4)x+cos^(4)x))/(sin^(3)x cos x)dx,x in(0,(pi)/(2))

Find the point of local maxima or local minima of the function f(x) = (sin^(4) x + cos^(4) x) " in " 0 lt x lt (pi)/(2)

If y=(sin^(4)x-cos^(4)x+sin^(2) x cos^(2)x)/(sin^(4) x+ cos^(4)x + sin^(2) x cos^(2)x), x in (0, pi/2) , then

Prove that : int_(0)^(pi//2) (x sin x cos x)/(sin^(4) x+ cos^(4)x)dx =(pi)/(16)