Question

Is `f(x) =cotx-cosx` concave or convex at `x=pi/3`?

Answer 1

`f(pi/3)>0=>"the function is concave at " x=pi/3`

Explanation:

• `f'(x)=-csc^2x+sinx`
  `f'(x)=0=>-csc^2x+sinx=0`
  `sinx=csc^2x`
  `sinx=1/sin^2x=>sin^3x=1`
  `sinx=1=>x=pi/2`
  :. the function has a local maxima/minima at `x=pi/2`

• `f(pi/3)=cot(pi/3)-cos(pi/3)=0.077`
  `f(pi/3)>0=>"the function is concave at " x=pi/3`

• If you check the graph
  `pi/2-=1.571=>pi/3-=1.05`
  graph{cotx-cosx [-0.706, 1.902, -0.256, 1.048]}