Question

How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for `y=sinx+x` for `[-pi,5pi]`?

Answer 1

Well, `sin(x)` is continuous on `RR`; so no worries about discontinuities.

Explanation:

As for finding points of inflection and concavity, we have to find the second derivative of your function and plug in values within your given interval.

`y' = cosx +1`

`y'' = -sinx`

Now that we have the second derivative, we have to find values of `x` that make `y'' = 0` or undefined.

Refer to the unit circle for values of `sinx=0`

We have `x=0, pi, 2pi, 3pi, . . .`

Thus we can write `x=pi+2kpi | kinZZ`

But we only want the values of `x in[-pi, 5pi]`

This includes `x= -pi, 0, pi, 2pi, 3pi, 4pi, & 5pi`

These values indicate you will have 6 intervals of concavity.

Plug values from each interval into your `f''` equation.

`[-pi, 0)` will be positive
`(0, pi)` will be negative
`(pi, 2pi)` will be positive
`(2pi, 3pi)` will be negative
`(3pi, 4pi)` will be positive
`(4pi, 5pi]` will be negative

Positive values indicate the function is concave UP on that interval.
Negative values indicate the function is concave DOWN on that interval.

The alternating (positive/negative signs) at each point indicate an inflection point at each value.

Answer 2

I am providing graph for the other answers as illustration for their findings. Note that `x=5pi=15.71`, nearly, and is included in the graph

Explanation:

You could spot easily all graphical properties including the wave formation, about the straight line y = x.

graph{y=x+sin x [-40, 40, -20, 20]}