Final answer:
To find the points of inflection for the function y = 3x^4 + 2x^3 - 5x - 12, we calculate the second derivative and set it to zero to find potential inflection points. We find two potential points where the concavity of the function may change. After testing the values around these points, we confirm there are two points of inflection.
Step-by-step explanation:
The question is asking to determine the number of points of inflection for the function y = 3x^4 + 2x^3 - 5x - 12. A point of inflection is where the second derivative changes sign, indicating a change in the concavity of the graph.
To find points of inflection, we need to:
- Find the second derivative of the function.
- Determine where the second derivative is zero or does not exist.
- Test the intervals around these values to see if the concavity changes.
First, the first derivative of the function is y' = 12x^3 + 6x^2 - 5. Next, the second derivative is y'' = 36x^2 + 12x.
We then set the second derivative equal to zero to find potential inflection points:
36x^2 + 12x = 0
x(36x + 12) = 0
x = 0 or x = -1/3
We now have two potential inflection points. Testing values around x = 0 and x = -1/3 will show a change in concavity, which can confirm these are indeed points of inflection.
Therefore, the function has two points of inflection, which corresponds to answer choice c.