The level of ozone, an invisible gas that irritates and impairs breathing, present in the atmosphere on a certain May day in the city was approximated by where is measured in pollutant standard index (PSI) and is measured in hours, with corresponding to a.m. Use the second derivative test to find the approximate value of where the function has a relative maximum. Round to the nearest integer.

Question

The level of ozone, an invisible gas that irritates and impairs breathing, present in the atmosphere on a certain May day in the city was approximated by  where  is measured in pollutant standard index (PSI) and  is measured in hours, with  corresponding to  a.m. Use the second derivative test to find the approximate value of  where the function  has a relative maximum. Round to the nearest integer.

Quizplus · Accepted Answer

To find the relative maximum, we need to look for the highest point on the curve. The second derivative test helps us to determine whether a critical point is a relative maximum, minimum, or a saddle point. 
Taking the second derivative of the given function, we get:
11eaa8b1_db7f_b9a9_b1b6_032043143290_TB6026_11 = 11eaa8b1_db7f_b9aa_b1b6_f7993a9dbc3a_TB6026_11 / (11eaa8b1_db7f_9294_b1b6_ad9ec21b30d2_TB6026_11)^2

Substituting 11eaa8b1_db7f_b9a6_b1b6_3543aa06250c_TB6026_11 in the equation, we get:
11eaa8b1_db7f_b9aa_b1b6_f7993a9dbc3a_TB6026_11 = 11eaa8b1_db7f_6b82_b1b6_f7a8930e2038_TB6026_11 × (1.0 / (11eaa8b1_db7f_9294_b1b6_ad9ec21b30d2_TB6026_11)^2)

At 11eaa8b1_db7f_b9a6_b1b6_3543aa06250c_TB6026_11, the value of the denominator is positive, so the sign of the second derivative is the same as the sign of the numerator. We need to determine the sign of the numerator to figure out if it is a relative maximum or not.

Since the value of the function is given in PSI and PSI is always non-negative, the second derivative must be non-negative as well. Therefore, 11eaa8b1_db7f_b9a9_b1b6_032043143290_TB6026_11 > 0, which means that the function has a relative minimum at this point. Hence, we need to look for the next highest point on the curve.

To find this point, we need to evaluate the second derivative at the nearby critical points until we find a negative value. Using the same method as above, we can evaluate the second derivative at 11eaa8b1_db7f_b9a8_b1b6_776584222983_TB6026_11 and get:
11eaa8b1_db7f_e0b9_b1b6_dff55d309051_TB6026_11 = 11eaa8b1_db7f_b9ab_b1b6_1b6c558fd097_TB6026_11 × (1.0 / (11eaa8b1_db7f_9295_b1b6_dbf465719ce7_TB6026_11)^2)
Substituting 11eaa8b1_db7f_b9a8_b1b6_776584222983_TB6026_11 and evaluating the second derivative we get,
11eaa8b1_db7f_b9ab_b1b6_1b6c558fd097_TB6026_11 = 11eaa8b1_db7f_6b82_b1b6_f7a8930e2038_TB6026_11 × (1.0 / (11eaa8b1_db7f_9295_b1b6_dbf465719ce7_TB6026_11)^2)
11eaa8b1_db7f_e0b9_b1b6_dff55d309051_TB6026_11 = 2.753

Therefore, the function has a relative maximum at 11eaa8b1_db7f_b9a8_b1b6_776584222983_TB6026_11, and the nearest integer is 3.

Hence, the correct choice is (C).

The Level of Ozone, an Invisible Gas That Irritates and Impairs