find profit

Question

john owns a hot dog stand.  his profit in dollars is given by the equation P(x)=P=-x^2+14x+54, where x is the number of hot dogs sold. what is the most he can earn?

Answer 1

okay, that's an upside-down U and the maximum value is at the top 

if you put the function in vertex form you can easily find the vertex:

group the first two terms and factor out the negative (essentially factoring out -1):

p = - (x²-14x) + 54

complete the square by taking the middle coefficient (14), dividing it by 2, squaring it and adding the result to the terms in parentheses

p = -(x²-14x+49) + 54 + 49  (we've really subtracted 49, so we have to undo this by adding 49)

p = -(x-7)² + 103

the vertex is at (7,103), so your maximum profit is $103

Answer 2

P(x)=P=x²+14x+54

Function y=ax²+bx+c is a parabola with vertex at x= -b/2a. It opens up if a>0 and it opens down if a <0.
So function P(x)=P=x²+14x+54 is a parabola with a=1, b=14, c=54. Since a=1 >0, it opens up.
Its vertex is at
x = -b/2a = -14/2 = -7
P(-7) = (-7)²+14(-7)+54 = 49-98+54 = $5
So the vertex is at (-7, 5)

P(-7) = $5 is the lowest profit point of the parabola since parabola opens up. From this point profit increases in either direction. But negative x values have no meaning in this application and when no hot dogs are sold (x=0), John makes a profit of P(0)=0²+14(0)+54=$54 !!!!

Since the parabola is opening up, you can see that as x (number of hotdogs) goes up, profit increases and there is no limit to the profit.

Second way to solve this problem is using derivatives.

This function will have maxima (highest profit) or minima (lowest profit) when its derivative = 0
dP/dx = 2x + 14 = 0
2x + 14 = 0
- 14 -14
2x = -14
x = -7
P(-7) = (-7)²+14(-7)+54 = 49-98+54 = $5
So the vertex is at (-7, 5)

The parabola opens up, if its second derivative is positive otherwise it opens down.
d²P/dx = d/dx (2x+14) = 2
Since second derivative is positive, parabola opens up so the first derivative gives us the minimum for profit of $5 and there is no limit to profit since profit increases as number of hot dogs sold increases.