Solve the problem ?

p '(x) = -2x + 14

0 = -2x + 14

x = 7

p ''(x) = -2 < 0 so we know a maximum occurs at x = 7.

maximum = p(7) = -7^2 + 14(7) + 58 = -49 + 98 + 58 = 107

p(x) = -x^2 +14x +58

p'(x) = -2x +14

x = 7 hotdogs

p(7) = -49 +98 +58 = $107

Best is to sell no hotdogs and collect $58. Do something else with your time.

p(x) = -x²+14x+58 is a CONCAVE DOWN parabola

..........(concave down, because the coefficient of the x²

..........term is negative; and parabola because the highest

..........exponent is 2)

The highest point of a concave down parabola is its vertex.

Given a parabola of the form f(x) = Ax²+Bx+C, the x-coordinate

of its vertex can be found at x = -B/2A.

p(x) has a vertex at (-14/(-2), p(7)) = (7, 107)

Therefore, the most John can earn is

$107, when he sells x=7 hotdogs......ANS

(see graph below)

boy would i like a hot dog. hotdogs are not in fashion and do not sell well, so i think he could only earn about 10 dolars if the hotdogs are a dollar a peice.

Use -b/2a from Quadratic Formula to fin min/max for x

x = -(-14)/2 = -7

then put x= 7 into formula to find y value.

Max when x = 7 is 107