7 - Economics uses

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Derivatives and their uses

Find the marginal functions:

\(TC(x) = x^{3} - 6x^{2} + 65x + 100\)
\(TR(x) = 100x - x^{2}\)
\(TR(x) = 75x\)
\(TU(x) = 10x - x^{2}\)
\(TP(K) = 60K^{2} - K^{3}\)

Marginal functions

\(MC(x) = 3x^{2} - 12x + 65\)
\(MR(x) = 100 - 2x\)
\(MR(x) = 75\)
\(MU(x) = 10 - 2x\)
\(MP(K) = 120K - 3K^{2}\)

Maximize the total revenue function: \(TR(x) = 40x - x^{2}\)

Critical point \(x = 20, {TR}''(2) = -2\), total revenue is maximized

Maximize the profit function: \(\pi (x) = - x^{2} + 12x - 30\)

Critical point \(x = 6, {\pi}''(6) = -2\), profit is maximized

Maximize the profit function: \(\pi (x) = -\frac{2}{3}x^{3} - 10x^{2} + 400x\)

Critical point \(x = -20, x = 10\), profit is maximized at \(x = 10\)

Maximize the profit function: \(\pi (x) = -\frac{2}{3}x^{3} - 10x^{2} + 400x - 100\)

Critical point \(x = -20, x = 10\), profit is maximized at \(x = 10\)

Find the critical value at which the average cost is minimized: \(TC(x) = x^{3} - 6x^{2} + 65x\)

Critical point \(x = 3, {AC}''(3) = 2\), \(AC\) is minimized

Find the critical value at which the marginal cost is the lowest: \(TC(x) = x^{3} - 6x^{2} + 65x\)

Critical point \(x = 2, {MC}''(2) = 6\), \(MC\) is minimized

Find the extreme value of the total cost function \(TC(x) = x^{3} - 5.5x^{2} + 150x + 675\)

The total cost function is always increasing; it has no extreme point