Let's analyze the graph of the function $f(x)$ and the statements:

Visual observations from the graph:

• f'(x) relates to the slope of the graph:

  • Where the function is increasing, $f'(x) > 0$.
  • Where the function is decreasing, $f'(x) < 0$.

• f''(x) relates to the concavity of the graph:

  • Where the graph is concave up, $f''(x) > 0$.
  • Where the graph is concave down, $f''(x) < 0$.

• The inflection points are where the concavity changes, typically seen as a change from concave up to concave down or vice versa.

Now let's address each condition in the problem:

(a) $f'(x) < 0$ for $0 < x < 2$

• From the graph, the function is decreasing between $x = 0$ and $x = 2$, meaning the slope $f'(x)$ is negative.
• True (T).

(b) $f'(x) > 0$ for $x > 2$

• The graph starts increasing after $x = 2$, meaning $f'(x) > 0$ for values greater than 2.
• True (T).

(c) $f''(x) < 0$ for $0 \leq x < 1$

• The function appears to be concave down between $x = 0$ and $x = 1$, meaning $f''(x) < 0$.
• True (T).

(d) $f''(x) > 0$ for $1 < x < 4$

• The graph changes from concave down to concave up after $x = 1$, meaning $f''(x) > 0$ in this range.
• True (T).

(e) $f''(x) < 0$ for $x > 4$

• The graph appears concave down for $x > 4$, so $f''(x) < 0$.
• True (T).

(f) Inflection points at $x = 1$ and $x = 3$

• Inflection points occur where the concavity changes. From the graph, it looks like the concavity changes at $x = 1$ and $x = 3$.
• True (T).

Would you like more details on any specific point or have further questions?

Related Questions:

1. What is the significance of $f'(x) = 0$ on the graph of $f(x)$?
2. How do inflection points relate to changes in concavity?
3. Can a point where $f'(x) = 0$ always be classified as a local minimum or maximum?
4. What happens to $f(x)$ when $f''(x) = 0$?
5. How do we identify local minima and maxima using the first derivative test?

Tip:

To identify where the concavity changes, look for points where the slope stops decreasing and starts increasing (or vice versa), as these are typically the inflection points.