Solve the equation P=100x-0.1x^2

Answer

$25,000

Q: Find the derivative of P to get equation with slope of the tangential line. A: dP/dx = 100 - 0.2x.
Q: Set the derivative equal to zero to find the critical points. A: 100 - 0.2x = 0 or x = 500.
Q: Find the second derivative of P to determine if the critical point is a maximum, minimum, or inflection point. A: d^2P/dx^2 = -0.2, which is negative, so x = 500 is a maximum.
Q: Substitute x into the original equation to find the maximum value of P. A: P = 100(500) - 0.1(500)^2 = $25,000.