Question

What are the points of inflection of `f(x)=x+cosx` on the interval `x in [0,2pi]`?

Answer 1

A point of inflection is a point in which the function switches from being concave to convex, or vice versa. Our tool to verify if a function is concave or convex is the second derivative, more precisely its sign: if `f''(x)>0` then `f` is concave in `x`, otherwise it's convex.

So, look for the convex/concave switch is the same as looking for the positive/negative switch for the second derivative.

So, first of all, let's compute it: since the derivative of a sum is the sum of the derivatives, we have

`f(x) = x + cos(x)`

`f'(x) = 1 - sin(x)`

`f''(x) = - cos(x)`

And since we need to refer to the `[0,2pi]` interval, the cosine is positive in `[0,pi/2]`, negative in `[pi/2,(3pi)/2]`, and again positive in `[(3pi)/2,2pi]`.

The second derivative is `-cos(x)`, so it will change negative areas with positive ones, but the switch points will still be `pi/2` and `(3pi)/2`