Show that the function is concave upward wherever it is defined.
A) The function g(x) is defined for any value of x and hence the function is concave upward for any x.
B) The second derivative of g(x) is . It is positive for any value of x, hence the function is concave upward for any x.
C) The first derivative of g(x) is . It is positive for any value of x, hence the function is concave upward for any x.

Question

Show that the function  is concave upward wherever it is defined.
A) The function g(x) is defined for any value of x and hence the function is concave upward for any x. 
B) The second derivative of g(x) is . It is positive for any value of x, hence the function is concave upward for any x. 
C) The first derivative of g(x) is . It is positive for any value of x, hence the function is concave upward for any x.

Quizplus · Accepted Answer

Concave upward means that the second derivative is positive. Option B shows that the second derivative of the function is positive for any value of x, which means the function is concave upward for any x. Option A is not entirely accurate because the fact that a function is defined for any value of x does not mean that it is necessarily concave upward. Option C is not enough to prove the concavity of the function, as having a positive first derivative only means that the function is increasing, not necessarily concave upward.

Show That the Function Is Concave Upward Wherever It