Find the smallest integer value of P for which the function `f(x)=6Px-P sin4x-5x-sin3x` is monotonic increasing and has no critical point on R

Find all the values of the parameter'a' for which the function; f(x)=8ax-a sin 6x-7x-sin 5x increases & has no critical points for all x in R .

Find the values of ' a ' for which the function f(x)=sinx-a x+4 is increasing function on R .

Find the values of ' a ' for which the function f(x)=sinx-a x+4 is increasing function on R .

The range of values of a for which the function f(x)=(a^2-7a+12)cosx+2(a-4)x+3e^5 does not process critical points is

The value of integer n for which the function f(x)=(sinx)/(sin(x / n) has 4pi its period is

Show that the function f(x) = x -sin x is an increasing function forall x in R .

The value of k for which the function f(x)={(sin(5x)/(3x)+cosx, xne0),(k, x=0):} is continuous at x=0 is

The length of the longest interval, in which the function 3sin x-4 sin^3x is increasing is

Find the set of all values of the parameter 'a' for which the function, f(x) = sin 2x - 8(a + b)sin x + (4a^2 + 8a - 14)x increases for all x in R and has no critical points for all a in R.

If f(x)=k x^3-9x^2+9x+3 monotonically increasing in R , then