Which of the following is a necessary condition for the inequality
[f(a) + f(b)] / 2 ≥ f((a + b) / 2)
to be true for all real numbers a and b with a < b?
Correct answer: C
Solution:
The expression [f(a) + f(b)] / 2 is the average of the two endpoint heights. Geometrically, it is the height of the midpoint of the straight line segment joining (a, f(a)) and (b, f(b)).
The expression f((a + b) / 2) is the height of the curve at the midpoint of the interval.
So the inequality says that the chord between two points on the graph lies above the graph at the midpoint. For this to happen for every interval, the graph must bend upwards, meaning the function must be convex.
For a polynomial function, this requires
f″(x) ≥ 0 for all real x.
The other options are not necessary. The function does not need to be positive, negative, increasing, or decreasing. What matters is the direction of curvature.
