Does anyone have any ideas on the following question?

Question

f(p) = 380p - 9p^3 - 860 + 18p^2

Find the value of p to obtain the maximum value of the above function.

Answer 1

There is no maximum value, as p approaches -infinity, the value of the function increases.

Answer 2

Assuming that this is in a calculus/business calc course...

Max and Min values are generally found using the first derivative. For this equation,
f'(p)=380-27p^2+36p
to make it easier, rewrite 380+36p-27p^2

Local (or relative) Max and min values occur where the first derivative =0, so

380+36p-27p^2=0 and solve for p
(I am lazy, so here are the solutions. . and they are UGLY)
(6-14sqrt6)/9 and (6+14sqrt6)/9
(decimal approximations are -3.144 and 4.477)

now, are we done? Nope. We are looking for a max, so we are looking for a place where the derivative changes from positive (original function increasing) to negative (original function decreasing)
so we have basically 3 zones to look at. . .numbers to the left of -3.144, numbers between -3.144 and 4.477, and finally numbers to the right of 4.77 (Choose EASY numbers to work with)
for numbers to the left of -3.144, let's stick in a -4 for p in the derivative. Doing the arithmetic, we get -196 (this indicates that the original function is DECREASING in areas to the left of -3.144).
now, test the middle area..an easy number between -3.144 and 4.477 is 0. Plug that in and we get 380. This indicates that the function is INCREASING between -3.144 and 4.477.

finally, test a number to the right of 4.477 How about 5? plug it in, clean it up, and we get -115. This indicates that the original function is again DECREASING to the righ of 4.477.

Since the function changes from Increasing to Decreasing (going up, then going back down), our local max occurs at p=4.477.

To find the actual value, plug that p in for the original function.

IF you are looking for an absolute max on this function, there is none (the range for a cube function is -infinity to +infinity.

Now, if you are doing this for an algebra level course, then stick the equation into your graphing calculator (replace p with x in the calculator), and set your window to "auto" (or whatever your model does to let you see the whole picture..I believe TI uses the "ZoomFit" command) Then push whatever buttons will give you the max for the area.

I hope this helps out!!

Answer 3

For what it's worth--the easiest way to find maximum p is to use Excel's Solver.
Enter in the target cell (say, A1) =380*A2 - 9*A2^3 - 860 + 18*a2^2
Press the Max button.
Solving, gives p =8.798+ in cell A2

Answer 4

Differentiate f(p), and solve for 0. The finite maxima is where p = (2/3) + (14/9)√6, or about 4.47698.

Answer 5

Take the derivative of f(p). At a maximum the derivative f '(p) equals zero. Plug zero in and solve for p.