One hour after x milligrams of a drug are given to a person, the change in body temperature T(x) (in degrees fahrenheit) is given by the function

T(x)=x^2(1−x/3).
Further, the rate T′(x) at which T changes with respect to the size of the dosage, x, is called the sensitivity of the body to the dosage. Find:
(A) T′(1) =

(B) T′(3) =

(C) T′(6) =

To find the sensitivity of the body to the dosage, we need to find T'(x), which represents the rate at which T (body temperature) changes with respect to the size of the dosage.

Given that T(x) = x^2(1 - x/3), we can find T'(x) by differentiating T(x) with respect to x.

Let's find T'(x):

Step 1: Expand T(x)
T(x) = x^2(1 - x/3)
= x^2 - (x^3)/3

Step 2: Differentiate T(x) with respect to x
T'(x) = d/dx (x^2 - (x^3)/3)
= 2x - (1/3)(3x^2)
= 2x - x^2

Now that we have T'(x), we can find the sensitivity of the body to the dosage at different values of x.

(A) T'(1):
Substitute x = 1 into T'(x):
T'(1) = 2(1) - (1)^2
= 2 - 1
= 1

Therefore, T'(1) = 1.

(B) T'(3):
Substitute x = 3 into T'(x):
T'(3) = 2(3) - (3)^2
= 6 - 9
= -3

Therefore, T'(3) = -3.

(C) T'(6):
Substitute x = 6 into T'(x):
T'(6) = 2(6) - (6)^2
= 12 - 36
= -24

Therefore, T'(6) = -24.

So, the answers are:
(A) T'(1) = 1
(B) T'(3) = -3
(C) T'(6) = -24