The worksheet contains problems related to logarithmic functions, focusing on analysis and interpretation. I'll help you tackle this step-by-step:

Part A

Function $g(x) = -x^2 + 1 - \ln x$

1. Study the variations of $g(x)$:

   • Find the derivative $g'(x)$: $g'(x) = -2x - \frac{1}{x}.$

   • Critical points: Solve $g'(x) = 0$: $-2x - \frac{1}{x} = 0 \implies -2x = \frac{1}{x} \implies -2x^2 = 1 \implies x = \pm \frac{1}{\sqrt{2}}.$ Since $x > 0$ (domain of $g(x)$), only $x = \frac{1}{\sqrt{2}}$ is valid.

   • Sign of $g'(x)$: Test intervals $(0, \frac{1}{\sqrt{2}})$ and $(\frac{1}{\sqrt{2}}, +\infty)$ for $g'(x)$:

     • For $x \in (0, \frac{1}{\sqrt{2}})$, $g'(x) > 0$ (increasing).
     • For $x \in (\frac{1}{\sqrt{2}}, +\infty)$, $g'(x) < 0$ (decreasing).

     Thus, $g(x)$ increases on $(0, \frac{1}{\sqrt{2}})$ and decreases on $(\frac{1}{\sqrt{2}}, +\infty)$.

2. Calculate $g(1)$ and study the sign of $g(x)$ on $(0, +\infty)$:

   • Compute $g(1)$: $g(1) = -1^2 + 1 - \ln 1 = 0.$
   • Sign of $g(x)$: Analyze $g(x) = -x^2 + 1 - \ln x$:
     • As $x \to 0^+$, $\ln x \to -\infty$, so $g(x) \to +\infty$.
     • As $x \to +\infty$, $-x^2$ dominates, so $g(x) \to -\infty$.

Part B

Function $f(x) = -\frac{1}{2}x + 1 + \frac{\ln x}{2x}$:

1. Study the limits of $f(x)$ at $0$ and $+\infty$:

   • As $x \to 0^+$: $f(x) = -\frac{1}{2}x + 1 + \frac{\ln x}{2x}.$ $\ln x \to -\infty$ dominates, so $f(x) \to -\infty$.

   • As $x \to +\infty$: $f(x) = -\frac{1}{2}x + 1 + \frac{\ln x}{2x}.$ The term $-\frac{1}{2}x$ dominates, so $f(x) \to -\infty$.

2. Find a relation between $f'(x)$ and $g(x)$:

   • Differentiate $f(x)$: $f'(x) = -\frac{1}{2} + \frac{1 - \ln x}{2x^2}.$
   • Rewrite $f'(x)$ to show its relationship with $g(x)$: Compare $f'(x)$ and $g(x) = -x^2 + 1 - \ln x$.

Would you like me to expand further on any of these points or calculations? Let me know if you'd like to see specific sections worked out in greater detail! Here are some related questions:

1. How do we determine the concavity of $g(x)$ or $f(x)$?
2. Can we find the exact coordinates of turning points for $g(x)$?
3. How does the asymptote of $f(x)$ influence the graph?
4. What do the roots of $f(x) = 0$ represent graphically?
5. How can we deduce the behavior of $f(x)$ near $x = 0$ and $+\infty$?

Tip: When studying variations, always check both the first derivative (for monotonicity) and the second derivative (for concavity) for a complete picture!