The question provided is in Hebrew and involves analyzing two graphs labeled I and II, where one represents a function $f(x)$ and the other its derivative $f'(x)$. It includes tasks such as identifying which graph corresponds to which function, determining critical points, and analyzing the derivative and second derivative for $f(x)$. Here's a structured response to the tasks based on the content of the problem:

Tasks Breakdown

Part A:

Identify which of the graphs corresponds to $f(x)$ and which to $f'(x)$. Justify the answer.

Part B:

1. Find the domain of $f(x)$.
2. Identify the critical points of $f(x)$ by analyzing $f'(x)$.
3. Determine the intervals where $f(x)$ is increasing or decreasing based on the derivative's sign.
4. Classify each critical point (minimum/maximum/inflection point) using the second derivative or the behavior of $f'(x)$.

Part C:

Given $g(x) = -f'(x)$, draw $g(x)$ based on the derivative graph $f'(x)$.

Solution Approach

Let’s address the tasks one by one:

Part A:

• Identifying $f(x)$ and $f'(x)$:
  • Graph I is the smoother curve that appears to describe the behavior of a function $f(x)$.
  • Graph II is the more jagged or oscillatory curve that seems to correspond to the derivative $f'(x)$, as it crosses the x-axis at the critical points of $f(x)$ (where $f'(x) = 0$).

Part B:

• 1. Domain of $f(x)$:

  • The function $f(x) = (x-5)^2 \cdot e^{-x} - 3$ is defined for all real numbers ($x \in \mathbb{R}$) because both $(x-5)^2$ and $e^{-x}$ are well-defined and continuous for all $x$.

• 2. Critical Points of $f(x)$:

  • Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined.
  • From graph II (the derivative), observe where it crosses the x-axis to identify the critical points.

• 3. Increasing/Decreasing Intervals:

  • $f(x)$ is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$.
  • Analyze the intervals on graph II (derivative).

• 4. Classify Critical Points:

  • Use the second derivative test ($f''(x)$) or examine the slope behavior (sign changes of $f'(x)$) to determine whether each critical point is a local maximum, minimum, or inflection point.

Part C:

• $g(x) = -f'(x)$:
  • The graph of $g(x)$ will be the inverse (reflection across the x-axis) of the derivative graph $f'(x)$ (Graph II). Carefully sketch $g(x)$ by flipping the values of $f'(x)$ vertically.

Let me know which part you'd like me to elaborate on, or if you'd like further clarifications!

Related Questions

1. What is the significance of critical points in the context of $f(x)$?
2. How do we use the derivative to classify increasing and decreasing intervals?
3. What role does the second derivative play in identifying concavity?
4. How is the graph of $g(x) = -f'(x)$ related to the graph of $f'(x)$?
5. Why is the domain of $f(x)$ important for analyzing its behavior?

Tip:

When analyzing graphs of functions and derivatives, always check where the derivative crosses the x-axis (critical points) and observe the sign changes to determine the function's behavior.