To analyze the function \( A(w) = w^2 + 5w \), we can determine its behavior over different intervals and identify any minimum or maximum values it may have.
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Determine the nature of the quadratic function: The equation \( A(w) = w^2 + 5w \) is a quadratic function in the standard form \( A(w) = aw^2 + bw + c \), where \( a = 1 \), \( b = 5 \), and \( c = 0 \). Since \( a > 0 \), this function opens upwards, indicating that it has a minimum point and no maximum values.
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Finding the vertex (minimum value): The vertex of a parabola represented by \( A(w) = aw^2 + bw + c \) occurs at \( w = -\frac{b}{2a} \). Here, \[ w = -\frac{5}{2 \cdot 1} = -\frac{5}{2} = -2.5. \] Since the vertex is at \( w = -2.5 \), and this value is not in the interval \( w > 0 \), it indicates the function keeps increasing for all \( w > 0 \).
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Behavior of \( A(w) \): For \( w > 0 \), since \( A(w) \) is a parabola that opens upwards, it will increase for \( w > -2.5 \) (which is always true for \( w > 0 \)). Therefore, \( A(w) \) increases as \( w \) increases for \( w > 0 \).
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Interval analysis: In the interval \( 0 < w < 5 \), since the function keeps increasing, we can confirm that \( A(w) \) increases in that range as well.
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Conclusion on options:
- Option A states that \( A(w) \) increases as \( w \) increases for \( 0 < w < 5 \): This is true.
- Option B states that \( A(w) \) increases as \( w \) increases when \( w > 0 \): This is also true.
- Option C states that \( A(w) \) has a minimum value at \( w = 5\): This is false; it has its minimum at \( w = -2.5\).
- Option D states that \( A(w) \) has a maximum value at \( w = 5\): This is also false; the function has no maximum as it continues to increase indefinitely.
Since both Options A and B are accurate, the most comprehensive and general statement about the behavior of \( A(w) \) is option B:
B. \( A(w) \) increases as \( w \) increases when \( w > 0 \).