Final answer:
The function f(x) is convex for all values of x greater than -1/2, which is the inflection point where the second derivative of f(x) is zero.
Step-by-step explanation:
The student's question asks about the concavity of the function f(x) = \frac{1}{3} x^3 + \frac{1}{2} x^2 - 6x + 2. The concavity of a function can be determined by computing the second derivative and finding where it is positive (convex) or negative (concave). To find the inflection point where the function changes concavity, we need to solve for when the second derivative is equal to zero.
In this case, the second derivative of f(x) is f''(x) = 2x + 1. Setting this equal to zero gives us x = -\frac{1}{2}, which would be the inflection point. For the function to be convex (where the second derivative is positive), we need all values of x greater than -\frac{1}{2}.