1. **Problem Statement:** We have two functions: $$f(n) = n \ln\left(\frac{10}{n}\right)$$ and $$h(n) = \frac{1}{\sqrt{n + c}}$$ where $n$ represents the original file size in MB, and $c$ is a constant. We want to analyze $f(n)$ on the interval $[1,10]$ to find where compression is most and least effective, i.e., find the extrema of $f(n)$ in this interval. 2. **Understanding the function $f(n)$:** - $f(n)$ is a product of $n$ and the natural logarithm of $\frac{10}{n}$. - The natural logarithm function $\ln(x)$ is defined for $x > 0$. - Since $n$ is in $[1,10]$, $\frac{10}{n}$ is positive, so $f(n)$ is well-defined. 3. **Finding critical points of $f(n)$:** We find where the derivative $f'(n)$ is zero or undefined to locate extrema. Using the product rule: $$f(n) = n \ln\left(\frac{10}{n}\right) = n (\ln 10 - \ln n) = n \ln 10 - n \ln n$$ Then, $$f'(n) = \frac{d}{dn}(n \ln 10) - \frac{d}{dn}(n \ln n) = \ln 10 - (\ln n + 1) = \ln 10 - \ln n - 1$$ Set $f'(n) = 0$: $$\ln 10 - \ln n - 1 = 0 \implies \ln n = \ln 10 - 1$$ Exponentiate both sides: $$n = e^{\ln 10 - 1} = 10 e^{-1} = \frac{10}{e} \approx 3.6788$$ 4. **Determine the nature of the critical point:** Calculate the second derivative: $$f''(n) = -\frac{1}{n}$$ At $n = \frac{10}{e}$, since $n > 0$, $f''(n) = -\frac{1}{n} < 0$, so $f(n)$ has a local maximum there. 5. **Evaluate $f(n)$ at the endpoints and critical point:** - At $n=1$: $$f(1) = 1 \times \ln(10/1) = \ln 10 \approx 2.3026$$ - At $n=10$: $$f(10) = 10 \times \ln(10/10) = 10 \times \ln 1 = 0$$ - At $n=\frac{10}{e}$: $$f\left(\frac{10}{e}\right) = \frac{10}{e} \times \ln\left(\frac{10}{\frac{10}{e}}\right) = \frac{10}{e} \times \ln e = \frac{10}{e} \times 1 = \frac{10}{e} \approx 3.6788$$ 6. **Interpretation:** - The maximum compression effectiveness (maximum $f(n)$) occurs at $n = \frac{10}{e} \approx 3.6788$ MB. - The least effective compression is at $n=10$ MB where $f(n)=0$. 7. **About $h(n) = \frac{1}{\sqrt{n + c}}$:** - This function decreases as $n$ increases because the denominator grows. - The constant $c$ shifts the function horizontally. - Without a specific $c$, we cannot analyze further. **Final answer:** - Maximum of $f(n)$ at $n = \frac{10}{e} \approx 3.6788$ MB. - Minimum of $f(n)$ at $n=10$ MB.