Second derivatives and bending

You will write a numerical second derivative, use its sign to say how a graph bends, and count how often the bend changes on an interval. The checker uses a quartic, a sine and two cubics.

The idea. The second derivative is the change of the slope. For a tiny $h$, $\dfrac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ estimates it. A positive value means the graph bends up, and a negative value means it bends down.

Your tasks.

1. Finish second_derivative(f, x). Return (f(x + H) - 2 * f(x) + f(x - H)) / (H * H).
2. Finish bend(f, x). Return "up" when the second derivative is above 0.001, "down" when it is below -0.001, and "neither" otherwise.
3. Finish count_bend_changes(f, a, b). Test 400 points between a and b, shifted half a step from the ends. Count the places where two neighbouring second derivatives have opposite signs.