Determine the signs (positive, negative, or zero) of
y

To determine the signs of $f(3)$, $f'(3)$, and $f''(3)$, let’s examine the graph and the characteristics of the function at $x = 3$.

1. Sign of $f(3)$:

   • $f(3)$ is the value of $f(x)$ at $x = 3$. Based on the graph, observe the $y$-value of $f(x)$ at $x = 3$. If the graph is above the $x$-axis, $f(3)$ is positive; if below, it’s negative; and if on the axis, it’s zero.

2. Sign of $f'(3)$:

   • $f'(3)$ represents the slope of $f(x)$ at $x = 3$. Look at the tangent at $x = 3$:
     • If the slope is positive (function increasing), $f'(3) > 0$.
     • If the slope is negative (function decreasing), $f'(3) < 0$.
     • If the slope is zero (horizontal tangent), $f'(3) = 0$.

3. Sign of $f''(3)$:

   • $f''(3)$ is the concavity or the rate of change of the slope at $x = 3$:
     • If the graph is concave up (curves upwards), $f''(3) > 0$.
     • If concave down (curves downwards), $f''(3) < 0$.
     • If there is no curvature (e.g., an inflection point), $f''(3) = 0$.

Based on your observations:

Select the answers for each sign based on these criteria.

---

Would you like more guidance on determining these values from a graph?

Related Questions

1. How can I find the critical points of a function from its graph?
2. How can I determine if a point is a local maximum or minimum using $f'(x)$ and $f''(x)$?
3. What is the importance of concavity in analyzing functions?
4. How can inflection points be identified from the graph?
5. How does the behavior of $f'(x)$ relate to the increase or decrease of $f(x)$?

Tip: For concavity, look at the "direction" the function curves — concave up resembles a "U" shape, while concave down resembles an "n" shape.