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6. Find a cubic function f(r) = ar³ + bx² + cx +d that has a local maximum value of 3 at r =-2 and a local minimum value of 0 at r = 1.

6. Find a cubic function f(r) = ar³ + bx² + cx +d that has a local maximum value of 3 at r =-2 and a local minimum value of 0 at r = 1.

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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How do I do number 6

### Mathematical Problems and Exercises #### 5 A number \( a \) is called a fixed point of a function \( f \) if \( f(a) = a \). Prove that if \( f'(x) \neq 1 \) for all real numbers \( x \), then \( f \) has at most one fixed point. #### 6 Find a cubic function \( f(x) = ax^3 + bx^2 + cx + d \) that has a local maximum value of 3 at \( x = -2 \) and a local minimum value of 0 at \( x = 1 \). #### 7 Sketch the graph of a function \( f \) that satisfies the following conditions: - \( f'(0) = f'(2) = f'(4) = 0 \) - \( f'(x) > 0 \) if \( x < 0 \) or \( 2 < x < 4 \) - \( f'(x) < 0 \) if \( 0 < x < 2 \) or \( x > 4 \) - \( f''(x) > 0 \) if \( 1 < x < 3 \) - \( f''(x) < 0 \) if \( x < 1 \) or \( x > 3 \) #### Evaluate the following limits: 8. \( \lim_{x \to -\infty} (x - \sqrt{x}) \) 9. \( \lim_{x \to \infty} \sqrt[3]{x} \sin\left(\frac{1}{x}\right) \) 10. \( \lim_{x \to 0^+} \frac{\frac{7}{x} + x + 5}{\sqrt{\frac{3}{x^2} + \frac{2}{x} + 1}} \)
Transcribed Image Text:### Mathematical Problems and Exercises #### 5 A number \( a \) is called a fixed point of a function \( f \) if \( f(a) = a \). Prove that if \( f'(x) \neq 1 \) for all real numbers \( x \), then \( f \) has at most one fixed point. #### 6 Find a cubic function \( f(x) = ax^3 + bx^2 + cx + d \) that has a local maximum value of 3 at \( x = -2 \) and a local minimum value of 0 at \( x = 1 \). #### 7 Sketch the graph of a function \( f \) that satisfies the following conditions: - \( f'(0) = f'(2) = f'(4) = 0 \) - \( f'(x) > 0 \) if \( x < 0 \) or \( 2 < x < 4 \) - \( f'(x) < 0 \) if \( 0 < x < 2 \) or \( x > 4 \) - \( f''(x) > 0 \) if \( 1 < x < 3 \) - \( f''(x) < 0 \) if \( x < 1 \) or \( x > 3 \) #### Evaluate the following limits: 8. \( \lim_{x \to -\infty} (x - \sqrt{x}) \) 9. \( \lim_{x \to \infty} \sqrt[3]{x} \sin\left(\frac{1}{x}\right) \) 10. \( \lim_{x \to 0^+} \frac{\frac{7}{x} + x + 5}{\sqrt{\frac{3}{x^2} + \frac{2}{x} + 1}} \)
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