Mathematics for Management -- Supplementary Electronic Materials

Quiz: Higher-Order Derivatives

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1. Find the second-order derivative of the function \(f(x) = x^4\) with respect to \(x\).

\( f''(x) = 12 x^2 \) \( f''(x) = 4 x^3 \) \( f''(x) = x^2 \) \( f''(x) = 4 x^2 \)

2. Find the second-order derivative of the function \(f(x) = 3 x^2 + 5x - 1\) with respect to \(x\).

\( f''(x) = 6x + 5 \) \( f''(x) = 5 \) \( f''(x) = 3x + 5 \) \( f''(x) = 6 \)

3. Find the second-order derivative of the function \(f(x) = 2x - 5x^6\) with respect to \(x\).

\( f''(x) = 2 - 30x \) \( f''(x) = -30 x^5 \) \( f''(x) = 2 - 20x^5 \) \( f''(x) = -150 x^4 \)

4. Find the second-order derivative of the function \(f(x) = \frac{3}{4} x^{1/2}\) with respect to \(x\).

\( f''(x) = \frac{3}{4} x{-3/2} \) \( f''(x) = -\frac{3}{16} x^{-1/2} \) \( f''(x) = \frac{1}{2} x^{-1/2} \) \( f''(x) = -\frac{3}{16} x^{-3/2} \)

5. Find the second-order derivative of the function \(f(x) = \frac{x^3 + x}{x^2}\) with respect to \(x\).

\( f''(x) = x + \frac{1}{x} \) \( f''(x) = \frac{2}{x^2} \) \( f''(x) = 1 - \frac{1}{x^2} \) \( f''(x) = \frac{2}{x^3} \)

6. Find the second-order derivative of the function \(f(x) = x^2 + {\rm{e}}^{x}\) with respect to \(x\).

\( f''(x) = 2 + {\rm{e}}^{x} \) \( f''(x) = 2x + x {\rm{e}}^{x} \) \( f''(x) = 2x + {\rm{e}}^{x} \) \( f''(x) = 2 {\rm{e}}^{x} \)

7. Find the third-order derivative of the function \(f(x) = 5x^2 (x + 47)\) with respect to \(x\).

\( f^{(3)}(x) = 15 \) \( f^{(3)}(x) = 15 + x \) \( f^{(3)}(x) = 30x \) none of these

8. Find \(f^{(3)}(x)\) for the function \(f(x) = 10 x^5 + 3 x^4 - 5 x^3 + 2 x^2 - 10 x + 6\).

\( f^{(3)}(x) = 200 x^3 + 36 x^2 - 30 x + 4 \) \( f^{(3)}(x) = 600 x^2 + 72 x - 30 \) \( f^{(3)}(x) = 600 x^2 + 72 x - 26 \) \( f^{(3)}(x) = 600 x^2 + 72 x - 36 \)

9. Let \(f(x) = \tfrac{21}{2} \cdot x^2 - 3{\rm{e}}^x\). Give the value of \(x\) for which the second derivative vanishes, i.e. \(f''(x)=0\).

\( \ln(7) \) \( 0 \) \( \frac{21}{6} \cdot {\rm{e}} \) \( {\rm{e}}^7 \)

10. Let \(u(x) = \sqrt{x}\) be a utility function. Find the ratio \[ -\frac{u''(x)}{u'(x)} \, , \] known as the Arrow-Pratt measure of relative risk aversion (you will encounter it in core microeconomics).

\( \frac{1}{2x} \) \( -\frac{1}{2x} \) \( 2x \) \( -2x \)

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Solution: 1a; 2d; 3d; 4d; 5d; 6a; 7d; 8b; 9a; 10a

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.