Explain why the equation cos x = x has at least one solution.

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Consider the function \( f(x) = \cos x - x \). We want to show that this function has at least one root, meaning there is at least one value of \( x \) for which \( f(x) = 0 \).

To apply the Intermediate Value Theorem, we need to find two values, \( a \) and \( b \), such that \( f(a) \) and \( f(b) \) have opposite signs. This will indicate that there is at least one root in the interval \([a, b]\).

Evaluate \( f(x) \) at \( x = 0 \): \( f(0) = \cos(0) - 0 = 1 \). So, \( f(0) = 1 \).

Evaluate \( f(x) \) at \( x = \pi/2 \): \( f(\pi/2) = \cos(\pi/2) - \pi/2 = 0 - \pi/2 = -\pi/2 \). So, \( f(\pi/2) < 0 \).

Since \( f(0) > 0 \) and \( f(\pi/2) < 0 \), by the Intermediate Value Theorem, there must be at least one value \( c \) in the interval \((0, \pi/2)\) such that \( f(c) = 0 \). Therefore, the equation \( \cos x = x \) has at least one solution.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function takes on two values at two points, it must also take on every value between those two points. This theorem is crucial for proving the existence of solutions to equations like cos x = x, as it guarantees that if the function changes sign over an interval, there is at least one root in that interval.

Continuity of Functions

A function is continuous if there are no breaks, jumps, or holes in its graph. The functions cos x and f(x) = x are both continuous over the real numbers. This property is essential for applying the Intermediate Value Theorem, as it ensures that the function behaves predictably and that solutions can be found within specified intervals.

Behavior of the Functions

To analyze the equation cos x = x, it's important to understand the behavior of the functions involved. The cosine function oscillates between -1 and 1, while the line y = x increases without bound. By examining the intersection points of these two graphs, we can conclude that there must be at least one point where they are equal, confirming the existence of a solution.