Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

g(t) = √t + √(1 + t) − 4, (0, ∞)

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First, understand that we need to show the function g(t) = √t + √(1 + t) − 4 has exactly one zero in the interval (0, ∞). This means we need to find where g(t) = 0 and prove it happens only once in the given interval.

To start, evaluate the behavior of g(t) as t approaches 0 and as t approaches infinity. As t approaches 0, g(t) approaches √0 + √1 - 4 = 1 - 4 = -3. As t approaches infinity, both √t and √(1 + t) grow without bound, so g(t) approaches infinity.

Next, check the continuity and differentiability of g(t) in the interval (0, ∞). Since both √t and √(1 + t) are continuous and differentiable for t > 0, g(t) is also continuous and differentiable in (0, ∞).

Apply the Intermediate Value Theorem. Since g(t) is continuous on (0, ∞) and changes from negative to positive as t increases from 0 to infinity, there must be at least one zero in the interval (0, ∞).

To show there is exactly one zero, consider the derivative g'(t) = (1/2√t) + (1/2√(1 + t)). Since both terms are positive for t > 0, g'(t) > 0 for all t in (0, ∞), indicating that g(t) is strictly increasing. Therefore, g(t) can cross the x-axis at most once, confirming exactly one zero in the interval.

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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, f, takes on values of opposite sign at two points, a and b, then it must have at least one root in the interval (a, b). This concept is crucial for proving the existence of a zero in a given interval by showing that the function changes sign.

Continuity of Functions

A function is continuous on an interval if it is continuous at every point within that interval. For the function g(t) = √t + √(1 + t) − 4, continuity is essential to apply the Intermediate Value Theorem. The square root functions involved are continuous for t > 0, ensuring g(t) is continuous on (0, ∞).

Behavior of Functions at Infinity

Understanding how a function behaves as t approaches infinity helps determine the number of zeros. For g(t) = √t + √(1 + t) − 4, as t increases, both √t and √(1 + t) grow, suggesting g(t) will eventually become positive. This behavior, combined with initial negative values, supports the existence of exactly one zero.

Roots (Zeros)Show that the functions in Exercises 19–26 have exactly one zero in the given interval.g(t) = √t + √(1 + t) − 4, (0, ∞)