Textbook Question
Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.
f"(x) > 0 on (-∞,-2); f"(-2) = 0; f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4,∞)
5. Graphical Applications of Derivatives
Curve Sketching
3:55 minutesProblem 4.4.8a
Textbook Question
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < -1
Verified step by step guidance1
Understand the given conditions: f'(x) < 0 implies the function f is decreasing, and f''(x) < 0 implies the function is concave down for x < -1.
For x < -1, since f'(x) < 0, the slope of the tangent line to the graph of f is negative, meaning the graph is sloping downward.
For x < -1, since f''(x) < 0, the graph is concave down, meaning it curves downward like an upside-down bowl.
Combine the two properties: The graph of f for x < -1 should be a decreasing curve that is concave down. This means the graph slopes downward and becomes steeper as x decreases.
Sketch the graph: Start at x = -1 and draw a curve that decreases and bends downward as x moves to the left. Ensure the curve reflects both the decreasing nature (f'(x) < 0) and the concave down property (f''(x) < 0).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First Derivative Test
The first derivative of a function, denoted as f', indicates the slope of the tangent line to the graph of the function at any point. If f' < 0, the function is decreasing in that interval. Understanding this concept is crucial for sketching the graph, as it informs us that the function is moving downward for x < -1.
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema
Second Derivative Test
The second derivative, denoted as f'', provides information about the concavity of the function. If f'' < 0, the function is concave down, meaning that the slope of the tangent line is decreasing. This concept is essential for understanding how the graph behaves in the specified interval, indicating that the function is not only decreasing but also bending downwards.
Recommended video:
06:02The Second Derivative Test: Finding Local Extrema
Graph Behavior
The overall behavior of a graph is influenced by both the first and second derivatives. In this case, since f' < 0 and f'' < 0 for x < -1, the graph will show a downward slope that becomes steeper as x decreases. Recognizing this behavior helps in accurately sketching the function, ensuring it reflects the properties of being decreasing and concave down.
Recommended video:
06:15Graphing The Derivative
11:41mWatch next
Master Summary of Curve Sketching with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice