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The graph of a function y = f(x) is shown below. Use this graph to draw a rough sketch of the graph y = f'(x).

The graph of a function y = f(x) is shown below. Use this graph to draw a rough sketch of the graph y = f'(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Graph of a Function and Its Derivative** The graph of a function \( y = f(x) \) is shown below. Use this graph to draw a rough sketch of the graph \( y = f'(x) \). **Graph Explanation:** - The graph depicts a smooth curve with two distinct peaks and one valley. - The curve starts from the top left, descends towards the x-axis forming a minimum point, rises to form the first peak, descends to form the valley, and finally rises to another peak, continuing upward. - The function has two maximum points (peaks) and one minimum point (valley). - At these extrema (maximum and minimum points), the derivative \( f'(x) \) will be zero because the slope of the tangent at these points is horizontal. **Sketching \( y = f'(x) \):** - Identify the x-coordinates where the original graph has peaks and valleys; these are where \( f'(x) = 0 \). - Determine the intervals where the function is increasing (positive slope) and decreasing (negative slope). - Between these critical points, sketch the derivative graph such that it crosses the x-axis at the critical points and has positive or negative values reflecting the increasing or decreasing nature of the function.
Transcribed Image Text:**Graph of a Function and Its Derivative** The graph of a function \( y = f(x) \) is shown below. Use this graph to draw a rough sketch of the graph \( y = f'(x) \). **Graph Explanation:** - The graph depicts a smooth curve with two distinct peaks and one valley. - The curve starts from the top left, descends towards the x-axis forming a minimum point, rises to form the first peak, descends to form the valley, and finally rises to another peak, continuing upward. - The function has two maximum points (peaks) and one minimum point (valley). - At these extrema (maximum and minimum points), the derivative \( f'(x) \) will be zero because the slope of the tangent at these points is horizontal. **Sketching \( y = f'(x) \):** - Identify the x-coordinates where the original graph has peaks and valleys; these are where \( f'(x) = 0 \). - Determine the intervals where the function is increasing (positive slope) and decreasing (negative slope). - Between these critical points, sketch the derivative graph such that it crosses the x-axis at the critical points and has positive or negative values reflecting the increasing or decreasing nature of the function.
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