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
Nov 22, 2016

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.

Nov 22, 2016

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]}