the drop in blood pressure of a typical patient who is given a certain medication is given by F(x)=0.028x^2(19-x) where x is the amount of medication in cubic centimeters. What is the maximum drop in blood pressure for this patient?
could use the product rule, but in this case I would expand
f(x) = .532x^2 - .028x^3
f'(x) = 1.064x - .084x^2
= 0 for a max/min
x(1.064 - .084x) = 0
x = 0, which will produce the min
or
x = 1.064/.084 = 12 2/3 = 38/3
f(38/3) = .... button - pushing time
(I got appr 28.45)
To find the maximum drop in blood pressure for this patient, we need to determine the maximum value of the function F(x)=0.028x^2(19-x).
One way to find the maximum value of a function is by finding the critical points. Critical points occur when the derivative of the function is either zero or undefined. Let's begin by finding the derivative of F(x):
F'(x) = d/dx (0.028x^2(19-x))
= 0.028 * [2x(19-x) + x^2]
Now, set the derivative equal to zero and solve for x to find the critical points:
0.028 * [2x(19-x) + x^2] = 0
Simplifying the equation:
2x(19-x) + x^2 = 0
2x(19-x) = -x^2
38x - 2x^2 = -x^2
2x^2 - 38x = 0
x(2x - 38) = 0
Now we have two possibilities:
1. x = 0
2. 2x - 38 = 0
Solving the second equation:
2x - 38 = 0
2x = 38
x = 38/2
x = 19
So, we have two critical points: x = 0 and x = 19.
Next, we need to evaluate the function F(x) at these critical points and determine which one gives the maximum value:
F(0) = 0.028(0)^2(19-0) = 0
F(19) = 0.028(19)^2(19-19) = 0
As we can see, both F(0) and F(19) are equal to 0. Therefore, there is no maximum drop in blood pressure for this patient according to the given function.
In other words, this function suggests that the blood pressure does not decrease or the decrease is negligible when given this medication.