# 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]?

Nov 22, 2016

Well, $\sin \left(x\right)$ is continuous on $\mathbb{R}$; 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.

$y ' = \cos x + 1$

$y ' ' = - \sin x$

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 $\sin x = 0$

We have $x = 0 , \pi , 2 \pi , 3 \pi , . . .$

Thus we can write $x = \pi + 2 k \pi | k \in \mathbb{Z}$

But we only want the values of $x \in \left[- \pi , 5 \pi\right]$

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.

$\left[- \pi , 0\right)$ will be positive
$\left(0 , \pi\right)$ will be negative
$\left(\pi , 2 \pi\right)$ will be positive
$\left(2 \pi , 3 \pi\right)$ will be negative
$\left(3 \pi , 4 \pi\right)$ will be positive
$\left(4 \pi , 5 \pi\right]$ 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.

Nov 22, 2016

I am providing graph for the other answers as illustration for their findings. Note that $x = 5 \pi = 15.71$, 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]}