# 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]#?

##### 2 Answers

Well,

#### 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.

Now that we have the second derivative, we have to find values of

Refer to the unit circle for values of

We have

Thus we can write

But we only want the values of

This includes

These values indicate you will have 6 intervals of concavity.

Plug values from each interval into your

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.

I am providing graph for the other answers as illustration for their findings. Note that

#### 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]}