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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.4.51
Chapter 4, Problem 4.4.51

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.


ƒ(x) = 3x⁴ + 4x³ - 12x²

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Step 1: Begin by identifying the key features of the function ƒ(x) = 3x⁴ + 4x³ - 12x². These include intercepts, local extreme values, and inflection points. To find the x-intercepts, set ƒ(x) = 0 and solve for x.
Step 2: Calculate the first derivative ƒ'(x) to find critical points, which are potential locations for local extreme values. The first derivative is given by ƒ'(x) = d/dx(3x⁴ + 4x³ - 12x²). Use the power rule to differentiate each term.
Step 3: Set the first derivative ƒ'(x) equal to zero and solve for x to find critical points. These points will help identify local maxima and minima. Also, consider the endpoints of the domain if applicable.
Step 4: Calculate the second derivative ƒ''(x) to determine concavity and locate inflection points. The second derivative is given by ƒ''(x) = d²/dx²(3x⁴ + 4x³ - 12x²). Again, use the power rule to differentiate.
Step 5: Use a graphing utility to plot the function ƒ(x) = 3x⁴ + 4x³ - 12x². Analyze the graph to confirm the locations of intercepts, local extreme values, and inflection points. Adjust the viewing window to ensure all relevant features are visible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Functions

Graphing functions involves plotting the curve of a mathematical function on a coordinate plane. This helps visualize the behavior of the function, including its intercepts, local extrema, and inflection points. Technology, such as graphing calculators or software, can assist in accurately plotting complex functions like polynomials.
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Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts occur where the function equals zero, and the y-intercept is where the function crosses the y-axis. Identifying intercepts is crucial for understanding the function's behavior and is often facilitated by graphing utilities.
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Slope-Intercept Form

Local Extrema and Inflection Points

Local extrema are points where a function reaches a local maximum or minimum. Inflection points are where the function's curvature changes direction. These points are critical for understanding the function's shape and behavior, and graphing technology can help locate them by analyzing the derivative and second derivative of the function.
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Critical Points
Related Practice
Textbook Question

Approximating changes


Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).

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Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = x ln x - 2x + 3 on (0,∞)

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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, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0

Textbook Question

{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.

Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?

Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 2x³ - 3x² + 12

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_u→ π/4 (tan u - cot u) / (u - π/4)