1. Concavity of a graph
A function is concave upward on an interval (a, b) if the graph of the function lies above its tangent lines at each point of (a, b). A function is concave downward on (a, b) if the graph of the function lies below its tangent line at each point (a, b). A point where a graph changes concavity is called a point of inflection.
Concave upward
tangent line slope zero
Refer to pages 757 and 758 of your text.
Concave downward
TEST FOR CONCAVITY
Let f be a function with derivatives f ' and f '' existing at all points in an interval (a, b). Then f is concave upward on (a, b) if f '' ( x ) > 0 for all x in (a, b), and concave downward if f '' ( x ) < 0 for all x in (a, b).

2. Point of Diminishing Returns
6. Find the point of diminishing returns (x, y) for the function, where R ( x ) represents revenue in thousands of dollars and x represents the amount spent on advertising in thousands of dollars.
Step 1. Find the first derivative.
Step 2. Find the second derivative.
Step 3. Set the second derivative equal to zero and solve.
– 4 x + 24 = 0
24 = 4 x
6 = x

3. This separates the problem into two interval (3, 6) and (6, 10).
Step 4. Substitute any value of x in the interval (3, 6) into the second derivative and evaluate. This answer must be positive if the point found in Step 3 is a point of diminishing return.
R '' ( 4 ) = – 4 ( 4 ) + 24 = – = 8
The positive 8 indicates the concavity is upward.
Step 5. Substitute any value of x in the interval (6, 10) into the second derivative and evaluate. This answer must be negative if the point found in Step 3 is a point of diminishing return.
R '' ( 8 ) = – 4 ( 8 ) + 24 = – = – 8
The negative 8 indicates the concavity is downward.

4. Step 6. Substitute the answer found in Step 3 into the original function and evaluate. Write answer in order pair form.
Answer: ( 6, 152 ) is the point of diminishing return.