onsider the profit function (Q is the quantity of goods sold)π = −12Q2 + 24Q + 53(a) Compute dπdQ and d2πdQ2 .(b) Use part (a) to find the maximum profit, and write the value of Q for whichthe maximum is attained
Question
onsider the profit function (Q is the quantity of goods sold)π = −12Q2 + 24Q + 53(a) Compute dπdQ and d2πdQ2 .(b) Use part (a) to find the maximum profit, and write the value of Q for whichthe maximum is attained
Solution
(a) To compute the first derivative of the profit function π with respect to Q (dπ/dQ), we apply the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1).
The derivative of -12Q^2 is -24Q (using the power rule, 2*-12Q^(2-1) = -24Q). The derivative of 24Q is 24 (since the derivative of a linear term is the coefficient of the variable). The derivative of 53 is 0 (since the derivative of a constant is zero).
So, dπ/dQ = -24Q + 24.
To compute the second derivative of the profit function π with respect to Q (d²π/dQ²), we differentiate the first derivative -24Q + 24 with respect to Q.
The derivative of -24Q is -24 (since the derivative of a linear term is the coefficient of the variable). The derivative of 24 is 0 (since the derivative of a constant is zero).
So, d²π/dQ² = -24.
(b) To find the maximum profit, we set the first derivative equal to zero and solve for Q.
-24Q + 24 = 0 -24Q = -24 Q = 1
The second derivative test tells us that if the second derivative at a point is negative, then the function has a local maximum at that point. Since d²π/dQ² = -24 is negative, we can conclude that the profit function has a local maximum at Q = 1.
Therefore, the maximum profit is attained when Q = 1. To find the maximum profit, we substitute Q = 1 into the profit function:
π = -12(1)^2 + 24(1) + 53 = -12 + 24 + 53 = 65.
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