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Chapter 4: Problem 7

Sketch a curve with the following properties. $$\begin{aligned} &f^{\prime}<0 \text { and } f^{\prime \prime}<0, \text { for } x<3\\\ &f^{\prime}<0 \text { and } f^{\prime \prime}>0, \text { for } x>3 \end{aligned}$$

Short Answer

Expert verified
Based on the given conditions, sketch the curve of the function by following these steps: 1. Begin by drawing the x-axis and labeling '3' on it. This is the point where the characteristics of the function shift. 2. For \(x<3\), draw a decreasing curve that is concave down. Start the curve fairly high above the x-axis to the left side of x=3. 3. For \(x>3\), draw a decreasing curve that is concave up. Begin the curve at x=3 and continue it to the right. 4. Connect the two parts of the curve by adjusting the concavity at x=3, making it a smoothly transitioning curve. 5. Verify that the properties of the curve match the given conditions: decreasing and concave down for x<3, and decreasing and concave up for x>3. Your sketch of the curve with the given properties is now complete.

Step by step solution

01

Draw the x-axis

Begin by drawing the x-axis and labeling '3' on it. This is the point where the characteristics of the function shift.
02

Determine the behavior for \(x

For \(x<3\), we know that the function is decreasing (since \(f'(x)<0\)) and concave down (since \(f''(x)<0\)). Draw a decreasing curve that changes concavity at x=3 (from concave down to up) as a reference point. It may be helpful to start the curve fairly high above the x-axis to the left side of x=3.
03

Determine the behavior for \(x>3\)

For \(x>3\), we know that the function is still decreasing (since \(f'(x)<0\)) but now the concavity changes, so it is concave up (since \(f''(x)>0\)). Begin drawing the curve starting at x=3 and continue it to the right with a decreasing pattern and a concave-up shape.
04

Connect the two parts

Now, you should have two parts of the curve: one to the left of x=3 that is decreasing and concave down, and one to the right of x=3 that is decreasing and concave up. Connect these two parts by adjusting the concavity of the curve at x=3.
05

Verify the properties of the curve

Finally, verify that the properties of the curve match the conditions given in the exercise. There should be a smoothly-transitioning curve that is decreasing and concave down for x<3, and decreasing and concave up for x>3. Now your sketch of the curve with the given properties is complete.

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