How do you find all points of inflection given #y=-(2x)/(x+3)#?

Question

Ezekiel Alexander · Accepted Answer

There aren't any?!

I may have gone wrong with this one but when I differentiated this and put #dy/dx = 0#, I got a quadratic which didn't factorise. I checked online with a graph plotter and the gradient of this graph has asymptotes so am I right in thinking there are no points of inflection?

Isaiah Allen · Answer

undefined To find all points of inflection for the function y = -(2x)/(x + 3), follow these steps:

1. Find the second derivative of the function.
2. Set the second derivative equal to zero and solve for x to find the potential points of inflection.
3. Test each potential point of inflection using the second derivative test.

First, find the second derivative:

y = -(2x)/(x + 3)
y' = -2(x + 3)^(-2)
y'' = 4(x + 3)^(-3)

Next, set the second derivative equal to zero:

4(x + 3)^(-3) = 0

The only solution to this equation is x = -3. This is the potential point of inflection.

To test this point, use the second derivative test:

Evaluate the second derivative at x = -3:

y''(-3) = 4(-3 + 3)^(-3)
        = 4(0)
        = 0

Since the second derivative is zero at x = -3, the second derivative test is inconclusive. Therefore, further analysis is needed.

To determine if x = -3 is a point of inflection, examine the behavior of the function around this point. You can do this by observing the concavity of the function on either side of x = -3. This can be done by examining the sign of the second derivative:

For x < -3, y'' is positive, indicating concave up.
For x > -3, y'' is negative, indicating concave down.

Since the concavity changes at x = -3, it is indeed a point of inflection.

Therefore, the point of inflection for the function y = -(2x)/(x + 3) is (-3, -3).