CMA Final · Strategic Cost Management · Business Application of Maxima and Minima
A firm's profit function is P(x) = -x³ + 12x² + 60x - 100 for x ≥ 0 (x in hundreds of units, P in Rs thousands). At which output does profit reach its local maximum?
Profit is maximised at x = 10 hundred units. Setting the first derivative -3x² + 24x + 60 to zero gives x = 10 as the only non-negative root, and the second derivative there is negative, confirming a maximum.
- Ax = 10Correct
- Bx = 6
- Cx = 12
- Dx = 5
Explanation
P'(x) = -3x² + 24x + 60 = 0, so x² - 8x - 20 = 0, giving (x-10)(x+2) = 0 and x = 10 (x = -2 rejected). P''(x) = -6x + 24 = -36 at x = 10, which is negative, so it is a maximum. x = 6 is where P'' = -12... not the root; x = 12 is the coefficient misused.
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