A manufacturer estimates that the profit from producing x units of a commodity is $- x ^ { 2 } + 40 x - 100$ dollars per week. What is the maximum profit he can realize in one week?
A) $300
B) $275
C) $500
D) $400
E) none of these

Question

Quizplus · Accepted Answer

The profit function is a downward-opening parabola, and its maximum value occurs at the vertex. The vertex of $y = -x^2 + 40x - 100$ is at $x = -b/(2a) = -40/(2 \times -1) = 20$. Substituting $x = 20$ into the profit function gives the maximum profit: $-20^2 + 40 \times 20 - 100 = 300$.

A Manufacturer Estimates That the Profit from Producing X Units −x2+40x−100- x ^ { 2 } + 40 x - 100−x2+40x−100

A Manufacturer Estimates That the Profit from Producing X Units $- x ^ { 2 } + 40 x - 100$