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 find the critical points first.

The first derivative is: 
11eb12b0_a9f2_ae5e_b394_9ad6df08e6f2_TB7866_11

Setting this equal to zero, we get: 
11eb12b0_a9f2_ae5f_b394_71b8a012f924_TB7866_11

So the critical point is at 11eb12b0_a9f2_d5a5_b394_5e5ec77f92d5_TB7866_11.

Now let's find the second derivative: 
11eb12b0_a9f2_ae60_b394_5d06c0ffd74a_TB7866_11

At 11eb12b0_a9f2_d5a5_b394_5e5ec77f92d5_TB7866_11, the second derivative is negative, so the point is concave down. Therefore, this critical point is a relative maximum.

Finally, we need to round 11eb12b0_a9f2_ae5c_b394_6dbfa82c3bd8_TB7866_11 to the nearest integer, which is 265.

Therefore, the answer is: 
Answer: C 
 The relative maximum of the function is at x = 265.

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