Question

Is `f(x) =sinx-x^2` concave or convex at `x=pi/3`?

Answer 1

Since the function is twice differentiable, the answer depends on whether the second derivative `f''(pi/3)` is positive or negative

Explanation:

Let's calculate the `f''(x)`. First `f'(x) = cosx-2x`, then `f''(x)=-sinx-2`. So, `f''(pi/3)=-sin(pi/3)-2`.

But `sin(pi/3)=sqrt3/2`, and then `f''(pi/3)=-sin(pi/3)-2=-sqrt3/2-2`, which is negative. Thus, since `f''(pi/3)<0`, the function is concave downward (convex)