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The function f(x) = 2x³ − 42x² + 270x – 4 has two critical numbers. The smaller one is x = and the larger one is x =

The function f(x) = 2x³ − 42x² + 270x – 4 has two critical numbers. The smaller one is x = and the larger one is x =

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 Critical Numbers of the Function** For the function \( f(x) = 2x^3 - 42x^2 + 270x - 4 \), we aim to find the critical numbers. Critical numbers are the values of \( x \) where the derivative of the function is zero or undefined. First, compute the derivative of the function: \[ f'(x) = \frac{d}{dx} (2x^3 - 42x^2 + 270x - 4) \] By applying the power rule: \[ f'(x) = 6x^2 - 84x + 270 \] Set the derivative equal to zero to find the critical numbers: \[ 6x^2 - 84x + 270 = 0 \] Solve this quadratic equation for \( x \): \[ x^2 - 14x + 45 = 0 \] Factor the quadratic equation: \[ (x - 9)(x - 5) = 0 \] This gives us two solutions: \[ x = 9 \text{ and } x = 5 \] Therefore, the function \( f(x) = 2x^3 - 42x^2 + 270x - 4 \) has two critical numbers. **Identifying the Critical Numbers:** 1. The smaller one is \( x = \) 5. 2. The larger one is \( x = \) 9.
Transcribed Image Text:**Finding Critical Numbers of the Function** For the function \( f(x) = 2x^3 - 42x^2 + 270x - 4 \), we aim to find the critical numbers. Critical numbers are the values of \( x \) where the derivative of the function is zero or undefined. First, compute the derivative of the function: \[ f'(x) = \frac{d}{dx} (2x^3 - 42x^2 + 270x - 4) \] By applying the power rule: \[ f'(x) = 6x^2 - 84x + 270 \] Set the derivative equal to zero to find the critical numbers: \[ 6x^2 - 84x + 270 = 0 \] Solve this quadratic equation for \( x \): \[ x^2 - 14x + 45 = 0 \] Factor the quadratic equation: \[ (x - 9)(x - 5) = 0 \] This gives us two solutions: \[ x = 9 \text{ and } x = 5 \] Therefore, the function \( f(x) = 2x^3 - 42x^2 + 270x - 4 \) has two critical numbers. **Identifying the Critical Numbers:** 1. The smaller one is \( x = \) 5. 2. The larger one is \( x = \) 9.
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