Let be a function defined for such that .How many inflection points does the graph of have on this interval?

Question

​Let  be a function defined for  such that  .How many inflection points does the graph of  have on this interval? ​

Quizplus · Accepted Answer

The given function is 11eac44f_ee14_9c80_bedf_a5f0ebd4742f_TB7497_11 and we are given that its second derivative is positive on the interval 11eac44f_ee14_9c81_bedf_df5bfef0a028_TB7497_11. This means that the graph of the function is concave up on that interval.

To find the inflection points of the graph, we need to look for points where the concavity changes, i.e., where the second derivative changes sign. Since the second derivative is positive on the entire interval, it can only change sign at points where it is zero. Therefore, we need to find the zeros of the second derivative.

The second derivative of the given function is 11eac44f_ee14_c394_bedf_582f757706e6_TB7497_11. Setting this equal to zero and solving for x, we get:

11eac44f_ee14_c395_bedf_aa6d4d6e2b9f_TB7497_11

This equation has four distinct real solutions, which means that the second derivative changes sign four times on the interval. Therefore, the graph of the function has four inflection points on the interval.

The answer is C.

​Let Be a Function Defined for Such That

Let Be a Function Defined for Such That