1. **Problem 3.1.1:** Find the absolute extreme values of $f(n) = n \ln(10n)$ on $[1,10]$. 2. **Step 1:** Compute the derivative $f'(n)$ to find critical points. $$f'(n) = \frac{d}{dn}[n \ln(10n)] = \ln(10n) + n \cdot \frac{1}{10n} \cdot 10 = \ln(10n) + 1$$ 3. **Step 2:** Set $f'(n) = 0$ to find critical points. $$\ln(10n) + 1 = 0 \implies \ln(10n) = -1 \implies 10n = e^{-1} \implies n = \frac{e^{-1}}{10} \approx 0.0368$$ 4. **Step 3:** Since $n=0.0368$ is not in $[1,10]$, no critical points inside the interval. 5. **Step 4:** Evaluate $f(n)$ at endpoints: - At $n=1$: $$f(1) = 1 \cdot \ln(10 \cdot 1) = \ln(10) \approx 2.3026$$ - At $n=10$: $$f(10) = 10 \cdot \ln(100) = 10 \cdot 4.6052 = 46.052$$ 6. **Step 5:** Determine absolute extrema: - Minimum at $n=1$ with $f(1) \approx 2.3026$ - Maximum at $n=10$ with $f(10) \approx 46.052$ --- 7. **Problem 3.1.2:** Why choose $[1,10]$ MB? 8. The interval $[1,10]$ MB likely represents practical file sizes for the study, focusing on typical small to medium files where compression efficiency is relevant and measurable. --- 9. **Problem 3.2:** Find critical points of $h(n) = \frac{n}{\sqrt{n} + c}$. 10. **Step 1:** Rewrite $h(n)$: $$h(n) = \frac{n}{n^{1/2} + c}$$ 11. **Step 2:** Use quotient rule: Let numerator $u = n$, denominator $v = n^{1/2} + c$. $$u' = 1, \quad v' = \frac{1}{2} n^{-1/2}$$ 12. **Step 3:** Derivative: $$h'(n) = \frac{u'v - uv'}{v^2} = \frac{(1)(n^{1/2} + c) - n \cdot \frac{1}{2} n^{-1/2}}{(n^{1/2} + c)^2} = \frac{n^{1/2} + c - \frac{1}{2} n^{1/2}}{(n^{1/2} + c)^2} = \frac{\frac{1}{2} n^{1/2} + c}{(n^{1/2} + c)^2}$$ 13. **Step 4:** Set $h'(n) = 0$: $$\frac{\frac{1}{2} n^{1/2} + c}{(n^{1/2} + c)^2} = 0 \implies \frac{1}{2} n^{1/2} + c = 0 \implies n^{1/2} = -2c$$ 14. **Step 5:** Since $n^{1/2} \geq 0$, for real $n$, $-2c \geq 0 \implies c \leq 0$. If $c \leq 0$, then critical point at: $$n = (-2c)^2 = 4c^2$$ If $c > 0$, no critical points. --- **Final answers:** - 3.1.1: Absolute minimum at $n=1$, $f(1) \approx 2.3026$; absolute maximum at $n=10$, $f(10) \approx 46.052$. - 3.1.2: Interval $[1,10]$ MB chosen for practical file size relevance. - 3.2: Critical points of $h(n)$ at $n=4c^2$ if $c \leq 0$; no critical points if $c > 0$.