bbolker
commented
6 years ago Can you be more explicit? The second derivative is everywhere negative -> the slope of the tangent is monotonically decreasing -> the function is decelerating -> the effect of Jensen's inequality is to reduce the mean of f(X) ; the second term in the approximation, 1/2 d^2f/dx^2 Var(x) is negative (because Var(x)>0).
What am I missing? The terminology of "concave" and "convex" is sometimes confusing (in some parts of the literature, people refer to "concave up" and "concave down".
duquea
The practice exam states that because the function has a negative second derivative at all points, that Jensen's inequality implies that the mean of the function is less than or equal to the function evaluated at the mean of x.
However, the fact that the second derivative of the function is negative at all points implies that the function is concave, not convex. Shouldn't the opposite hold?