Question

How do you find all points of inflection given `y=-2sinx`?

Answer 1

There is a point of inflection whenever `-2sinx=0`

Explanation:

Points of inflection occur when the curve changes concavity. Since this is a sine wave, there are an infinite number of points of inflection.

A function is concave up when the second derivative (`f''`) is greater than 0, and concave down when the second derivative is below 0. Critical points, therefore, are when the second derivative equals 0.

Differentiate `y=-2sinx` to get `y'=-2cosx`. Differentiate again to get `y''=2sinx`, the original function.

Whenever `-2sinx=0`, there is a point of inflection. This can be intuitively verified by graphing `y=-2sinx`.

graph{-2sinx [-pi, pi, -3, 3]}

Whenever the curve crosses the x-axis (that is, whenever y=0), the concavity changes.