Find the Inflection Points y=x-sin(x)

Problem

y=x−sin(x)

Solution

1. Find the first derivative of the function to begin the process of finding the second derivative.

d(y)/d(x)=1−cos(x)

2. Find the second derivative by differentiating the first derivative with respect to x

d2(y)/(d(x)2)=sin(x)

3. Set the second derivative to zero to find potential inflection points.

sin(x)=0

4. Solve for x to identify the points where the concavity might change.

x=n*π

where n is any integer.

5. Verify the change in concavity by checking the sign of the second derivative on either side of x=n*π Since sin(x) changes sign at every multiple of π these are all inflection points.

6. Find the y-coordinates by substituting the x values back into the original function y=x−sin(x)

y=n*π−sin(n*π)

y=n*π−0

y=n*π

Final Answer

Inflection Points: *(n*π,n*π)* for any integer *n

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