Question

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

Answer 1

Increasing. See the illustrative Socratic graph for the point
`(pi/3, sqrt3-sqrt3/3)=(1.047, 0.8660)`, nearly, at which f is increasing.

Explanation:

`f'=sec^2x-cosx`.

#f'(pi/3)=sec^(pi/3)-cos(pi/3)=(2)^2-1/2=3.5>0.

So, f is increasing at `x = pi/3`.

graph{(y-tanx + sin x) (y-0.8660) = 0 [-1.6, 1.6, -1.5, 1.5]}