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 downward

Concave upward

Refer to pages 757 and 758 of your text.

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).

tangent line slope zero

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.

10 3 ,142512)( 2332 xxxxxR

Step 1. Find the first derivative.

25242)(' 2 xxxR

Step 2. Find the second derivative.

Step 3. Set the second derivative equal to zero and solve.

244)('' xxR

0244)('' xxR

– 4 x + 24 = 0

24 = 4 x

6 = x

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 = – 16 + 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 = – 32 + 24 = – 8

The negative 8 indicates the concavity is downward.

Step 6. Substitute the answer found in Step 3 into the original function and evaluate. Write answer in order pair form.

15214)6(25)6(12)6()6( 2332 R

Answer: ( 6, 152 ) is the point of diminishing return.