Determine the points of inflection of the function.

Question

Determine the points of inflection of the function.

f(x) = x + sin(x)    (−2π < x < 2π)

(x,y)=( , ) smallest x value

(x,y)=( , )

(x,y)=( , ) largest x value

Answer 1

Inflection points occur wherever the second derivative changes from positive to negative. This can happen wherever the second derivative equals zero (f '' = 0) or the second derivative is undefined (f '' = und).

To solve this problem we need to find the second derivative first.

The first derivative is: f ' = 1 + cos(x)

The second derivative is: f '' = -sin(x)

This means that wherever -sin(x) = 0 there COULD be an inflection point. We are given the domain of

[-2π x < 2π]. There are three x values that make -sin(x) = 0 in that domain. Once you have those three values plug them into the original f(x) function to get there corresponding y values.

Edit: You should do a number line test with the three x values. Verify that the sign of f '' = -sinx actually does change as it crosses those three values.

Answer 2

y' = 1 + cos x = 0

cos x = -1

x = pi