Example 5 In the year 2 0 1 1 , a group of musicians launched a surf music social website, a site on which people who enjoy listening to instrumental surf guitar could communicate and interact with each other The site receives revenue from organizations that place advertisements on the site 's pages The musicians constructed the revenue function R ( x ) 1 4 x 4 2 0 0 x 2 to model the revenue R ( x ) the site should realize from online advertising x months after the beginning of 2 0 1 1 A financial analyst claims that under this model, the site will realize its maximum revenue 3 0 months after the beginning of 2 0 1 1 and then from that time on the revenue will decrease slowly to 0 Assuming the revenue model to be true, should the website 's managers accept or reject this analyst 's forecast Solution The analyst makes statements about a function 's maximum value and its decreasing at a decreasing rate behavior Such a claim involves the first and second derivatives Begin by finding the first and second derivative of R ( x ) and the critical values R ' ( x ) x 3 4 0 0 x and R ' ' ( x ) 3 x 2 4 0 0 R ' ( x ) 0 when x 3 4 0 0 x 0 Solving this equation, we get x ( x 2 4 0 0 ) 0 x ( x 2 0 ) ( x 2 0 ) 0 x 0 , 2 0 , 2 0 The values 0 and 2 0 make no sense in the context of the problem, so the only critical value is 2 0 We can make a sign chart to see where the function is increasing or decreasing Figure 1 0 As the sign chart shows, at x 2 0 , R ( x ) changes from increasing to decreasing, indicating a maximum occurs at that point This means the media site will see its maximum revenue 2 0 months after the beginning of 2 0 1 1 , contradicting the analyst 's claim of 3 0 months Also for all values of x 2 0 , R ' ( x ) 0 and R ' ' ( x ) 0 These results mean that for all months after the maximum revenue is realized, the revenue function decreases at an increasing rate This, again, contradicts the analyst 's forecast that the site 's revenue will decrease slowly to 0 The revenue will decrease quickly to 0 The website 's managers should reject both the analyst 's forecasts decreases at an increasing rate This, again, contradicts the analyst 's forecast that the site 's revenue will decrease slowly to 0 The revenue will decrease quickly to 0 The website 's managers should reject both the analyst 's forecasts In Section 3 1 , we used the first derivative test to determine if a critical value of a function f produced a relative maximum or a relative minimum Because a function is concave upward at a point if its second derivative is positive at that point, it makes sense that if x c is a critical value of the function, then it must produce a relative minimum ( See Figure 1 1 ) Likewise, because a function is concave downward at a point if its second derivative is negative at that point, it makes sense that if x c is a critical value of the function, then it must produce a relative maximum See Figure 1 1 x c produces a relative minimum and fis concave x c produces a relative upward at x 6 downward at x c Figure 1 1 5 Repeat example 5 using the function R ( t ) 1 4 x 4 4 5 0 x 2 Show all images Show all images Show all images done loading

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Example 5 In the year 2 0 1 1 , a group of musicians launched a surf music social website, a site on which people who enjoy listening to instrumental surf guitar could communicate and interact with each other  The site receives revenue from organizations that place advertisements on the site 's pages  The musicians constructed the revenue function R ( x )     1 4 x 4 2 0 0 x 2 to model the revenue R ( x ) the site should realize from online advertising x months after the beginning of 2 0 1 1   A financial analyst claims that under this model, the site will realize its maximum revenue 3 0 months after the beginning of 2 0 1 1 and then from that time on the revenue will decrease slowly to 0   Assuming the revenue model to be true, should the website 's managers accept or reject this analyst 's forecast  Solution The analyst makes statements about a function 's maximum value and its decreasing   at   a   decreasing   rate behavior  Such a claim involves the first and second derivatives  Begin by finding the first and second derivative of R ( x ) and the critical values  R  ' ( x )     x 3 4 0 0 x and R  '  ' ( x )     3 x 2 4 0 0 R  ' ( x )   0 when   x 3 4 0 0 x   0 Solving this equation, we get   x ( x 2   4 0 0 )   0   x ( x 2 0 ) ( x   2 0 )   0 x   0 ,   2 0 , 2 0 The values 0 and   2 0 make no sense in the context of the problem, so the only critical value is 2 0   We can make a sign chart to see where the function is increasing or decreasing  Figure 1 0 As the sign chart shows, at x   2 0 , R ( x ) changes from increasing to decreasing, indicating a maximum occurs at that point  This means the media site will see its maximum revenue 2 0 months after the beginning of 2 0 1 1 , contradicting the analyst 's claim of 3 0 months  Also for all values of x   2 0 , R  ' ( x ) 0 and R  '  ' ( x ) 0   These results mean that for all months after the maximum revenue is realized, the revenue function decreases at an increasing rate  This, again, contradicts the analyst 's forecast that the site 's revenue will decrease slowly to 0   The revenue will decrease quickly to 0   The website 's managers should reject both the analyst 's forecasts decreases at an increasing rate  This, again, contradicts the analyst 's forecast that the site 's revenue will decrease slowly to 0   The revenue will decrease quickly to 0   The website 's managers should reject both the analyst 's forecasts  In Section 3   1 , we used the first derivative test to determine if a critical value of a function f produced a relative maximum or a relative minimum  Because a function is concave upward at a point if its second derivative is positive at that point, it makes sense that if x   c is a critical value of the function, then it must produce a relative minimum  ( See Figure 1 1   ) Likewise, because a function is concave downward at a point if its second derivative is negative at that point, it makes sense that if x   c is a critical value of the function, then it must produce a relative maximum  See Figure 1 1   x   c produces a relative minimum and fis concave x   c produces a relative upward at x   6 downward at x   c Figure 1 1 5   Repeat example 5 using the function R ( t )     1 4 x 4 4 5 0 x 2 Show all images Show all images Show all images done loading

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Example 5 In the year 2 0 1 1 , a group of musicians launched a surf music social website, a site on which people

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