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Math M1314
lesson 1314 Lesson 11 Applications of the Second Derivative Concavity Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change of the function is increasing or decreasing. In business, for example, the first derivative might tell us that our profits are increasing, but the second derivative will tell us if the pace of the increase is increasing or decreasing. Example 1:
From these graphs, you can see that the shape of the curve change differs depending on whether the slopes of tangent lines are increasing or decreasing. This is the idea of concavity. Definition: Let the function f be differentiable on an interval (a, b). Then f is concave upward on (a, b) if f ' is increasing on (a, b) and f is concave downward on (a, b) if f ' is decreasing on (a, b).
Determining Where a Function is Concave Upward and Where it is Concave Downward By Analyzing the Sign of the Second Derivative Algebraically We can also determine concavity algebraically. The procedure for doing this should look pretty familiar:
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1. Find the second derivative of the function. 2. Determine all values of x for which f ( x ) = 0 or is undefined. 3. Use the values found in step 2 to divide the number line into open intervals. 4. Choose a test value, c, in each open interval. 5. Substitute each test value, c, into the second derivative to determine the sign of f (c) . 6. Apply the following theorem: Theorem: If f ' ' ( x ) > 0 for each value of x in (a, b), then f is concave upward on (a, b). If f ' ' ( x ) < 0 for each value of x in (a, b), then f is concave downward on (a, b). Example 1: Determine where the function f ( x ) = x 3 - 3x 2 - 24 x + 32 is concave upward and where it is concave downward.
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Example 2: Determine where the function f ( x ) = xe 2 x is concave upward and where it is concave downward.
Example 3: Determine where the function f ( x ) = x 4 + 6 x 3 is concave upward and where it is concave downward.
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Inflection Points The point where a function changes from being concave upward to concave downward (or from being concave downward to concave upward) is called a point of inflection or an inflection point. We'll show the significance of this point by an example. Example 4:
Note that, at the beginning, the slopes of the lines tangent to the graph are increasing slowly. Then the slopes of the tangent lines increase rapidly. But after a certain point, the rate of increase slows down. Suppose you are in business and this is a graph of your total sales, where x represents the amount spent (in millions of dollars) on advertising and y represents total receipts (in millions of dollars). When your company an begins advertising campaign, the rate of increase in sales speeds up. But this can't go on forever. There comes a point where the advertising campaign won't have as much of an effect on sales. This is the point of inflection. You may think of it as the point of diminishing returns. In business, it might be the time to start a new ad campaign.
M1314
Definition: A point on the graph for a differentiable function f at which the concavity changes is called an inflection point. Finding Inflection Points To find the inflection point(s) of a function,
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1. Compute f ' ' ( x ) . 2. Find all points in the domain of f for which f ' ' ( x ) = 0 . 3. Determine the sign of f ' ' ( x ) to the left and to the right of each point x = c found in step 2. If there is a change in the sign of f ' ' ( x ) as we move across the point x = c from left to right, then the point (c , f (c)) is an inflection point of f.
Note: to find the y coordinate of an inflection point, substitute the x coordinate into the original function. Example 5: Determine any points of inflection if f (x) = x 4 - 2 x 3 + 6 .
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Example 6: Determine any points of inflection if f ( x ) = xe 2 x .
Example 7: Determine any points of inflection if f (x) = x 4 - 5x 3 + 2 x + 3 .
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The Second Derivative Test Recall that we use the first derivative test to determine if a critical point is a relative extremum. There is also a second derivative test to find relative extrema. It is sometimes convenient to use; however, it can be inconclusive. The Second Derivative Test: 1. Find f ' ( x ) and f ' ' ( x ) . 2. Find all critical points. 3. Compute f ' ' (c) fo...