29. Sketch the graph of a function F for which \( f(-2)=5, f^{\prime \prime}(-2)=0, f(0)=0 \), and \( f^{\prime \prime}(4)=3 \)

To sketch a graph of a function \( f(x) \) that satisfies the given conditions: 1. **Given Conditions:** - \( f(-2) = 5 \): The function passes through the point \((-2, 5)\). - \( f''(-2) = 0 \): The second derivative at \( x = -2 \) is zero, indicating a possible inflection point. - \( f(0) = 0 \): The function passes through the origin \((0, 0)\). - \( f''(4) = 3 \): The second derivative at \( x = 4 \) is positive, indicating the graph is concave upward at this point. 2. **Interpreting the Conditions:** - **Point \((-2, 5)\):** Since \( f''(-2) = 0 \), this point is likely an inflection point where the concavity of the function changes. - **Point \((0, 0)\):** The function crosses the origin. - **Point \((4, \text{unknown})\):** At \( x = 4 \), the function is concave upward because \( f''(4) = 3 > 0 \). 3. **Sketching the Graph:** Here's a step-by-step guide to sketching such a function: - **Plot the Points:** - Mark the point \((-2, 5)\). - Mark the origin \((0, 0)\). - **Determine Concavity:** - **For \( x < -2 \):** Choose a concavity (e.g., concave down). - **At \( x = -2 \):** Since \( f''(-2) = 0 \), an inflection point occurs here. If the function was concave down before \( x = -2 \), it changes to concave up after \( x = -2 \). - **For \( -2 < x < 4 \):** Maintain the new concavity (e.g., concave up). - **At \( x = 4 \):** The function remains concave up since \( f''(4) = 3 \). - **Draw the Curve:** - Start from the left, ensuring the curve approaches \((-2, 5)\) with the initial concavity. - Pass through \((-2, 5)\), changing concavity at this point. - Curve smoothly towards the origin \((0, 0)\) while maintaining concave up. - Continue the curve past the origin, ensuring that by \( x = 4 \), the concavity remains upward. 4. **Sample Sketch:** While I can't provide a visual sketch here, here's a description of what the graph might look like: - **Left of \( x = -2 \):** The graph is concave down, increasing towards \((-2, 5)\). - **At \( x = -2 \):** An inflection point where the concavity changes from down to up. - **Between \( x = -2 \) and \( x = 0 \):** The graph is concave up, decreasing from \((-2, 5)\) to \((0, 0)\). - **Between \( x = 0 \) and \( x = 4 \):** The graph continues to be concave up, curving smoothly and perhaps increasing as it approaches \( x = 4 \) with \( f''(4) = 3 \). - **At \( x = 4 \):** The graph is concave upward with a positive second derivative. 5. **Additional Considerations:** - **Smoothness:** Ensure the function is smooth (differentiable) at all points, especially at the inflection point \( x = -2 \). - **Behavior at Infinity:** Depending on further requirements, you can decide how the function behaves as \( x \) approaches positive or negative infinity (e.g., tends to infinity, approaches horizontal asymptotes, etc.). 6. **Example Function:** To provide a concrete example, consider a polynomial function that meets the conditions. For instance, a cubic or quartic polynomial could be designed to satisfy the given values and concavity conditions. Here's an illustrative example using a cubic function: \[ f(x) = ax^3 + bx^2 + cx + d \] By plugging in the conditions: - \( f(-2) = 5 \) - \( f(0) = 0 \) implies \( d = 0 \) - \( f''(x) = 6ax + 2b \) - \( f''(-2) = 0 \) implies \( 6a(-2) + 2b = 0 \) → \( -12a + 2b = 0 \) → \( b = 6a \) - \( f''(4) = 3 \) implies \( 6a(4) + 2b = 3 \) → \( 24a + 12a = 3 \) → \( 36a = 3 \) → \( a = \frac{1}{12} \), \( b = \frac{1}{2} \) Now, the function is: \[ f(x) = \frac{1}{12}x^3 + \frac{1}{2}x^2 + cx \] Use \( f(-2) = 5 \) to find \( c \): \[ \frac{1}{12}(-2)^3 + \frac{1}{2}(-2)^2 + c(-2) = 5 \\ \frac{1}{12}(-8) + \frac{1}{2}(4) - 2c = 5 \\ -\frac{2}{3} + 2 - 2c = 5 \\ \frac{4}{3} - 2c = 5 \\ -2c = 5 - \frac{4}{3} = \frac{11}{3} \\ c = -\frac{11}{6} \] So, an example function is: \[ f(x) = \frac{1}{12}x^3 + \frac{1}{2}x^2 - \frac{11}{6}x \] **Graphing this Function:** - **At \( x = -2 \):** \( f(-2) = 5 \), and the graph has an inflection point. - **At \( x = 0 \):** \( f(0) = 0 \). - **At \( x = 4 \):** \( f''(4) = 3 \), confirming concave upward. This specific function meets all the given conditions and provides a concrete example for the sketch. ## Final Sketch Description 1. **Plot Key Points:** - \((-2, 5)\) - \((0, 0)\) 2. **Inflection Point at \( x = -2 \):** - Concave changes from down to up. 3. **Concave Upward at \( x = 4 \):** - Ensure the graph is concave upward in this region. 4. **Smooth Curve:** - Connect the points with a smooth curve respecting the concavity changes. By following these guidelines, you'll have a graph that satisfies all the given conditions.