Final answer:
To determine the number of points of inflection for f(x) = 3x^4 + 2x^3 - 5x - 12, calculate the second derivative and find where it changes sign. After finding the second derivative is zero at x = 0 and x = -1/3, checking around these points for a sign change will confirm any points of inflection.
Step-by-step explanation:
To find the number of points of inflection for the function f(x) = 3x^4 + 2x^3 - 5x - 12, we need to investigate the concavity changes of the function. A point of inflection occurs where the second derivative changes sign, indicating a change in concavity.
First, compute the first derivative of f(x):
f'(x) = d/dx (3x^4 + 2x^3 - 5x - 12) = 12x^3 + 6x^2 - 5
Next, compute the second derivative: f''(x) = d/dx (12x^3 + 6x^2 - 5) = 36x^2 + 12x
Then, find the values of x where f''(x) is zero to determine possible points of inflection:
- Set f''(x) = 0
- 0 = 36x^2 + 12x
- x(36x + 12) = 0
- x = 0 or x = -1/3