I need help, can anyone answer this?

Question

The total profit in thousands of dollars from the sale of x thousand units of a new arthritis drug is give by the function: f(x)= -x^3+7.5x^2+72x. How many units should be sold to maximize the company’s profit? Will the company have a point where the problem will reach its lowest value?

 

Answer 1

Brian,

 

In order to find the minimum or maximum points, we need to first take the first derivative of the function f(x) and set it equal to zero to find the critical values for x.  We then take the second derivative to determine whether these critical points are minimum or maximum.

 

The Second Derivative Test states:

     When a function's slope is zero (first derivative) at x, and the second derivative at x is:
          -  less than 0, it is a local maximum
          -  greater than 0, it is a local minimum
          -  equal to 0, then the test fails (there may be other ways of finding out though)

 

So:

 

f(x) = -x^3 + 7.5x^2 + 72x

f'(x) = -3x^2 + 15x + 72 = 0

 

Use the quadratic formula to solve for x:

 

x=(- b +/- sqrt(b^2-4ac))/2a

 

x = (-15 +/- sqrt((15*15) + 4(-3)(72)))/(2(-3)

x= (-15 +/- sqrt(225 + 864))/-6

x = 7.4 or -2.4

 

 

However, x cannot be negative so the only answer is x = 7.4.

 

Now let's use the second derivative test to determine whether x=7.4 is a maximum or minimum.

 

f'(x) = -3x^2 + 15x + 72

f''(x) = -6x + 15

 

Substitute x = 7.4 into the equation:

 

f''(x) = -6(7.4) + 15 = -29.4.

 

According to the Second Derivative Test, since f''(7.4) is negative, then we have a local maximum and therefore will maximize profits.

 

Answer is x = 7.4 or 7400 units to maximize profits.

 

 

In order to answer the second part of the question, we need to find the values of x where y = 0.

 

f(x) = -x^3 + 7.5x^2 + 72x

f(x) = -x(x^2 - 7.5x -72) = 0

 

The values of x where y = 0  are x = -5.5, 0 or 13.0.  Use the quadratic formula to determine -5.5 and 13.0.  Again, since x cannot be negative, the lowest values for x are x=0 and x = 13.0.

 

Hope this helps!

 

 

 

 

Answer 2

Brian,

            The function has local maximum at x=8 or 8000 units. It has non negative roots at x=0 and x=13.07 these are the lowest values where the function is still >0.

 

Jim