Question

The function f is continuous on the interval (0,16), and f is twice differentiable except at x=5 where the derivatives are undefined. Information about the first and second derivatives of f for values of x in the interval (0,16) is given in the table above. At what values of x in the interval (0,16) does the graph of f have a point of inflection?.

Answer 1

Using it's definition, it is found that the function f(x) has a point of inflection at:

A. x = 8 only.

What are the points of inflection of a function?

• The critical points of a function are the values of x for which:

$f^{\prime\prime}(x) = 0$

• Additionally, there has to be a change in the sign of $f^{\prime\prime}(x)$

Researching the problem on the internet, it is found that:

• For 0 < x < 5, $f^{\prime\prime}(x) > 0$.

• For x = 5, $f^{\prime\prime}(x)$ is undefined.

• For 5 < x < 8, $f^{\prime\prime}(x) < 0$.

• For x = 8, $f^{\prime\prime}(x) = 0$.

• For 8 < x < 12, $f^{\prime\prime}(x) > 0$.

• For x = 12, $f^{\prime\prime}(x) = 0$.

• For 12 < x < 16, $f^{\prime\prime}(x) > 0$.

The two conditions, $f^{\prime\prime}(x) = 0$ and a change in the signal of $f^{\prime\prime}(x)$ are only respected at x = 8, which is the lone inflection point.

You can learn more about points of inflection at brainly.com/question/10352137