Question

Is `f(x)=cscx-sinx` increasing or decreasing at `x=pi/3`?

Answer 1

`f` is `darr` at `x=pi/3`.

Explanation:

A function `f` is `uarr or darr "at" x=a, "according as" f'(a)> or < 0`.

Now, `f(x)=csc x-sin x rArr f'(x)=-csc xcot x-cos x=-(csc xcot x+cos x)`

`:. AA x in (0,pi/2), f'(x)<0, as, AA x in (0,pi/2), (csc xcot x+cos x)>0`.

Therefore, `f` is `darr AA x In (o,pi/2)`.

In particular, `f` is `darr` at `x=pi/3`.