![Rolle's Theorem for f(x) = x^2 in [-1, 1]](https://images.unsplash.com/photo-1570914755800-cc174d78aa6b?crop=entropy&cs=srgb&fm=jpg&ixid=M3w0MjA4MDF8MHwxfHNlYXJjaHwxfHxSb2xsZSUyN3MlMjBUaGVvcmVtJTJDJTIwY2FsY3VsdXMlMkMlMjBtYXRoZW1hdGljc3xlbnwxfDB8fHwxNzI3MTAzNzY3fDA&ixlib=rb-4.0.3&q=85&w=800)
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Questions and Answers
What is the derivative of the function f(x) = x^2?
What is the derivative of the function f(x) = x^2?
Which conditions must be satisfied to apply Rolle's theorem on the interval [-1, 1] for f(x) = x^2?
Which conditions must be satisfied to apply Rolle's theorem on the interval [-1, 1] for f(x) = x^2?
What is the value of f(-1) and f(1) for the function f(x) = x^2?
What is the value of f(-1) and f(1) for the function f(x) = x^2?
Where does the derivative of f(x) = x^2 equal zero in the interval [-1, 1]?
Where does the derivative of f(x) = x^2 equal zero in the interval [-1, 1]?
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Which of the following best describes the nature of the function f(x) = x^2 in the interval [-1, 1]?
Which of the following best describes the nature of the function f(x) = x^2 in the interval [-1, 1]?
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Which of the following correctly explains the continuity of f(x) = x^2 on the interval [-1, 1]?
Which of the following correctly explains the continuity of f(x) = x^2 on the interval [-1, 1]?
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What does Rolle's theorem guarantee for the function f(x) = x^2 on the interval [-1, 1]?
What does Rolle's theorem guarantee for the function f(x) = x^2 on the interval [-1, 1]?
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If Rolle's theorem applies to f(x) = x^2 in the interval [-1, 1], what can you say about the function's behavior in that interval?
If Rolle's theorem applies to f(x) = x^2 in the interval [-1, 1], what can you say about the function's behavior in that interval?
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When evaluating f(x) = x^2 at the endpoints of [-1, 1], what value is obtained?
When evaluating f(x) = x^2 at the endpoints of [-1, 1], what value is obtained?
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What is the derivative of the function f(x) = x^2 used to apply Rolle's theorem?
What is the derivative of the function f(x) = x^2 used to apply Rolle's theorem?
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Study Notes
Derivative and Evaluation
- The derivative of the function f(x) = x² is f'(x) = 2x.
- Evaluating the function at the endpoints, f(-1) = (-1)² = 1 and f(1) = 1² = 1.
Rolle's Theorem Conditions
- For Rolle's theorem to apply, the function must be continuous on the closed interval [-1, 1] and differentiable on the open interval (-1, 1).
- f(x) = x² is continuous everywhere and differentiable everywhere.
Application of Rolle's Theorem
- The conditions are met because f(-1) = f(1), both equal to 1.
- Rolle's theorem guarantees at least one c in (-1, 1) where f'(c) = 0.
Derivative and Critical Points
- Setting f'(x) = 0 gives 2x = 0, leading to x = 0 as the only critical point in the interval [-1, 1].
Nature of the Function
- The function f(x) = x² is a continuous and differentiable quadratic function, always non-negative, with a minimum at the vertex (0, 0) in the interval.
Theorem Implications
- Since the conditions of Rolle's theorem hold, f(x) = x² shows a smooth transition from 1 to 1 across [-1, 1] with a critical point at x = 0.
- The function is symmetric about the y-axis and opens upwards, indicating the behavior is parabolic and concave up throughout the interval.
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Description
This quiz explores the application of Rolle's Theorem to the function f(x) = x^2 over the interval [-1, 1]. It examines the function's derivative, boundary values, and the conditions necessary for the theorem to hold. Test your understanding of critical points and the implications of Rolle's Theorem in this specified interval.
