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Locate the absolute extrema of the function on the closed interval. g(x) = x² - 2x, [0, 9] minimum (x, y) = (1,-1 maximum (x, y) = (DNE ) × )

Locate the absolute extrema of the function on the closed interval. g(x) = x² - 2x, [0, 9] minimum (x, y) = (1,-1 maximum (x, y) = (DNE ) × )

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Finding Absolute Extrema of a Function on a Closed Interval** Consider the function and interval given below: \[ g(x) = x^2 - 2x, \quad \text{for } x \in [0, 9] \] To find the absolute extrema (both minima and maxima) of the function \( g(x) \) in the interval \([0, 9]\): 1. **Calculate the Critical Points**: - Derivative of \( g(x) \): \( g'(x) = 2x - 2 \) - Set \( g'(x) = 0 \) to find critical points: \( 2x - 2 = 0 \implies x = 1 \) 2. **Evaluate \( g(x) \) at Critical Points and Endpoints**: - \( g(0) = 0^2 - 2 \cdot 0 = 0 \) - \( g(1) = 1^2 - 2 \cdot 1 = -1 \) - \( g(9) = 9^2 - 2 \cdot 9 = 81 - 18 = 63 \) 3. **Determine Extrema**: - Absolute minimum value: \( g(1) = -1 \) - Absolute maximum value: \( g(9) = 63 \) However, it appears from the image text that there was a mistake: - **Minimum**: Correctly identified at \( (1, -1) \) - **Maximum**: Incorrectly indicated as "DNE" (Does Not Exist), but the actual maximum value should occur at point \( (9, 63) \). ### Summary: - **Minimum (x, y) = (1, -1)** ✔️ - **Maximum (x, y) = (9, 63)** It's important to correctly verify all endpoints and critical points within the given interval to ensure the determination of absolute extrema.
Transcribed Image Text:**Finding Absolute Extrema of a Function on a Closed Interval** Consider the function and interval given below: \[ g(x) = x^2 - 2x, \quad \text{for } x \in [0, 9] \] To find the absolute extrema (both minima and maxima) of the function \( g(x) \) in the interval \([0, 9]\): 1. **Calculate the Critical Points**: - Derivative of \( g(x) \): \( g'(x) = 2x - 2 \) - Set \( g'(x) = 0 \) to find critical points: \( 2x - 2 = 0 \implies x = 1 \) 2. **Evaluate \( g(x) \) at Critical Points and Endpoints**: - \( g(0) = 0^2 - 2 \cdot 0 = 0 \) - \( g(1) = 1^2 - 2 \cdot 1 = -1 \) - \( g(9) = 9^2 - 2 \cdot 9 = 81 - 18 = 63 \) 3. **Determine Extrema**: - Absolute minimum value: \( g(1) = -1 \) - Absolute maximum value: \( g(9) = 63 \) However, it appears from the image text that there was a mistake: - **Minimum**: Correctly identified at \( (1, -1) \) - **Maximum**: Incorrectly indicated as "DNE" (Does Not Exist), but the actual maximum value should occur at point \( (9, 63) \). ### Summary: - **Minimum (x, y) = (1, -1)** ✔️ - **Maximum (x, y) = (9, 63)** It's important to correctly verify all endpoints and critical points within the given interval to ensure the determination of absolute extrema.
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