Optimizing a Continuous Function on a Closed Interval

00:00 6 marks

A manufacturer models the profit (in dollars) from producing a product by the function \(f(x) = -2x^2 + 20x + 100\), where \(x\) (in thousands) represents the number of units produced. The production is limited to the interval \([0, 12]\).

(a) Find the stationary point(s) of \(f(x)\) within the interval.

(b) Evaluate \(f(x)\) at the stationary point and at the endpoints \(x = 0\) and \(x = 12\).

(c) Determine the maximum profit and justify why checking the endpoints is necessary to find the absolute maximum profit in this production scenario, given the production is limited to a closed interval.

Hint Hint

Hint

Differentiate the function, set the derivative equal to zero to find stationary points, and then evaluate the function at the stationary point and both endpoints.

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Answer Answer

Complete Answer

The stationary point is at \(x = 5\) with \(f(5) = 150\). Since \(f(0) = 100\) and \(f(12) = 52\), the maximum profit is \(150\) dollars at \(x = 5\).

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