Find the points of inflection and discuss the concavity of the function on the interval .
A) concave down on ; no points of inflection
B) concave downward on ; concave upward on ; inflection points at and
C) concave upward on ; concave downward on ; inflection points at and
D) concave up on ; no points of inflection
E) none of the above

Question

Find the points of inflection and discuss the concavity of the function  on the interval  .
A) concave down on ; no points of inflection 
B) concave downward on ; concave upward on ; inflection points at and 
C) concave upward on ; concave downward on ; inflection points at and 
D) concave up on ; no points of inflection 
E) none of the above

Quizplus · Accepted Answer

To find the points of inflection, we need to find where the second derivative changes sign. 
f(x) = 11e^(x/2) - 9e^(-x/2) 
f'(x) = (11/2)e^(x/2) + (9/2)e^(-x/2) 
f''(x) = (11/4)e^(x/2) - (9/4)e^(-x/2) 
For concavity to change from upward to downward, we need f''(x) to change from positive to negative. This occurs at x = ln(9/11), which is in the given interval. For concavity to change back from downward to upward, we need f''(x) to change from negative to positive. This occurs at x = ln(11/9), also in the given interval. Therefore, the function is concave downward on (ln(9/11), ln(11/9)) and concave upward elsewhere. The inflection points are (ln(9/11), f(ln(9/11))) and (ln(11/9), f(ln(11/9))).

Find the Points of Inflection and Discuss the Concavity of the Function