In a certain biological models, the human body's reaction to...

In a certain biological models, the human body's reaction to a particular kind of medication is measured by a function of the form F(M) = 1/3(kM^2 - M^3), 0 ? k ? M where k is a positive constant and M is the amount of medication absorbed in the blood? The sensitivity of the body to the medication is measured by the derivative S = F'(M). a. Show that the body is most sensitive to the medication when M = k/3. b. What is the average reaction to the medication for 0 ? M ? k/3.

Transcript

00:01 Hi, i'm david and i'm here to help me answer your question.

00:04 Now let me bring up your question here.

00:07 In the question we're given the function f as a function of the variable m.

00:15 And then here we are given that the derivative f prime of m.

00:22 It will be the sensitivity on the body to the medication is measured by the derivative.

00:29 They want to show that the body is more sensitive.

00:31 To the medication when k equal to 103.

00:35 So in the to do that, we have to find the f -bramm and then we will have the derivative equal to the one of the three and we have the two km minus 3m square.

00:58 And then we set the f -bram equal to 0.

01:01 Then we serve for the k.

01:05 So this implies that we have the 2k m must equal to the 3m about 2.

01:15 And therefore we should get the k it will equal to the 3m square over the 2m.

01:26 Then we get equal to the 3m over 2.

01:34 And then the next time we need to find the second derivative here and for the second derivative we have the 1 out of 3 and then we have the 2k minus 6m equal to 0 and from here we should be able to find the value of the k and we have the 2k minus 6m equal to 0 and then the k is equal to the 6m over 2 and equal to the 3m.

02:41 So m is equal to the 2k over 6th.

02:50 So this one will equal to the k over 3.

02:56 And okay, so this one we have 2 so for the m, not for the k, so from here, we should get the m, it will equal to the 2k over 6, and then we can equal to k out of 3.

03:12 And this will be the second derivative equal to 0.

03:17 So because the sensitivity of the body, it will be the derivative of the s, derivative of the m.

03:26 So therefore, we want to find the most sensitivity...

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