Determine the signs (positive, negative, or zero) of  
y
=
f
(
x
)
  (shown in the graph),  
f
'
(
x
)
 , and  
f
'
'
(
x
)
  when x = 3.

The sign of  
f
(
3
)
  is 
Select an answer

The sign of  
f
'
(
3
)
  is 
Select an answer

The sign of  
f
'
'
(
3
)
  is 
Select an answer


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.

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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.

Determine the signs (positive, negative, or zero) of y = f(x) (shown in the graph), f'(x), and f''(x) when x = 3.

The sign of f(3) is Select an answer
The sign of f'(3) is Select an answer
The sign of f''(3) is Select an answer

Learn how to determine the signs of f(x), f'(x), and f''(x) at x = 3 using graph analysis. This solution explains how to identify function values, slopes, and concavity at specific points, suitable for high school students studying calculus concepts.

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

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