# What are the points of inflection of #f(x)= (x^2 - 8)/(x+3) #?

There are no points of inflections

We need

Here,

Consequently, the initial derivative is

Furthermore, the second derivative is

graph{[-58.53, 58.55, -29.24, 29.23]} (x^2-8)/(x+3)

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To find the points of inflection, first find the second derivative of f(x), then solve for the values of x where the second derivative equals zero or is undefined. Then, determine if the concavity changes at those points by examining the sign of the second derivative around those values of x. In this case, the second derivative is (18 - 2x)/(x + 3)^3. Setting this equal to zero and solving for x gives x = 9. Since the second derivative is undefined at x = -3, we have two potential points of inflection: x = -3 and x = 9. To determine if they are points of inflection, check the concavity around these points. You can use the first or second derivative test for concavity.

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