Question

Suppose the function f has a second derivative f''(x)=x²(x-6)⁴(x-4)³(x²-1). How many inflection points does f have? Where do they occur?

Answer 1

To determine the number of inflection points of a function, analyze the behavior of the second derivative, which represents the concavity. Inflection points occur where the concavity changes.

Step-by-step explanation:

To determine the number of inflection points of the function f, we need to analyze the behavior of the second derivative f''(x). Inflection points occur at values of x where the concavity of the graph changes. The second derivative tells us the concavity of the function.

Given that f''(x) = x²(x-6)⁴(x-4)³(x²-1), we know that the concavity of the function changes at x-values where f''(x) = 0 or where there are vertical tangents at x-values where f''(x) is undefined.

Therefore, to find the inflection points, we need to find the x-values that make f''(x) = 0 or are undefined.