Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Concept explainers

Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
Question
**Problem Statement:** The relative maximum value of \( f(x) = x^2 \cdot \sqrt[3]{x - 7} \) is __________, and its relative minimum value is __________. **Explanation:** In this problem, we are given a function \( f(x) = x^2 \cdot \sqrt[3]{x - 7} \) and asked to find its relative maximum and minimum values. To solve this, we can follow these steps: 1. **Find the first derivative:** Compute \( f'(x) \) to determine where the slope is zero or undefined. These points are potential candidates for relative maxima and minima. 2. **Find critical points:** Solve \( f'(x) = 0 \) for \( x \). These solutions are the critical points. 3. **Second derivative test:** Compute \( f''(x) \). Use the second derivative test to identify if each critical point is a relative maximum or minimum. 4. **Evaluate the function at critical points:** Substitute the critical points back into \( f(x) \) to find the values of the relative maximum and minimum. By following these steps, you can determine the relative extrema of the function. (Note: In a classroom or educational setting, you can include intermediate steps for calculations to aid students' understanding.)
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Transcribed Image Text:**Problem Statement:** The relative maximum value of \( f(x) = x^2 \cdot \sqrt[3]{x - 7} \) is __________, and its relative minimum value is __________. **Explanation:** In this problem, we are given a function \( f(x) = x^2 \cdot \sqrt[3]{x - 7} \) and asked to find its relative maximum and minimum values. To solve this, we can follow these steps: 1. **Find the first derivative:** Compute \( f'(x) \) to determine where the slope is zero or undefined. These points are potential candidates for relative maxima and minima. 2. **Find critical points:** Solve \( f'(x) = 0 \) for \( x \). These solutions are the critical points. 3. **Second derivative test:** Compute \( f''(x) \). Use the second derivative test to identify if each critical point is a relative maximum or minimum. 4. **Evaluate the function at critical points:** Substitute the critical points back into \( f(x) \) to find the values of the relative maximum and minimum. By following these steps, you can determine the relative extrema of the function. (Note: In a classroom or educational setting, you can include intermediate steps for calculations to aid students' understanding.)
The image contains text from an educational setting, likely a multiple-choice question related to calculus or related fields of mathematics. The text can be transcribed as follows for an educational website: --- **Question:** If \( c \) is a critical value of a function \( f \), then which of the following statements must be true? - A. \( f(c) = 0 \) - B. \( f'(c) = 0 \) - C. none of these - D. \( f\) is undefined - E. \( f'(c) \) d.n.e. --- This question tests the understanding of critical values in the context of calculus. It is crucial to remember that a critical value occurs where the derivative of a function is zero or undefined.
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Transcribed Image Text:The image contains text from an educational setting, likely a multiple-choice question related to calculus or related fields of mathematics. The text can be transcribed as follows for an educational website: --- **Question:** If \( c \) is a critical value of a function \( f \), then which of the following statements must be true? - A. \( f(c) = 0 \) - B. \( f'(c) = 0 \) - C. none of these - D. \( f\) is undefined - E. \( f'(c) \) d.n.e. --- This question tests the understanding of critical values in the context of calculus. It is crucial to remember that a critical value occurs where the derivative of a function is zero or undefined.
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