Find the largest interval on which the given function is increasing.

d. R(x) = √ 2x - 1

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First, understand that a function is increasing on an interval if its derivative is positive on that interval. So, we need to find the derivative of the function R(x) = √(2x - 1).

To find the derivative, use the chain rule. The function can be rewritten as (2x - 1)^(1/2). The derivative of this function is (1/2)(2x - 1)^(-1/2) * (d/dx)(2x - 1).

Calculate the derivative of the inner function (2x - 1), which is simply 2. Therefore, the derivative of R(x) is (1/2)(2x - 1)^(-1/2) * 2.

Simplify the expression for the derivative: R'(x) = (2/2)(2x - 1)^(-1/2) = (2x - 1)^(-1/2).

Determine where the derivative is positive. Since (2x - 1)^(-1/2) is positive when 2x - 1 > 0, solve the inequality 2x - 1 > 0 to find the interval where the function is increasing.

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주요 개념

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

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. To determine where a function is increasing, we analyze its derivative: if the derivative is positive over an interval, the function is increasing on that interval.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for identifying intervals of increase or decrease, as they can indicate potential local maxima or minima, which help in determining the overall behavior of the function.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It is crucial for expressing the intervals on which a function is increasing or decreasing. For example, the interval (a, b) indicates that the function is increasing from point a to point b, not including the endpoints.

Find the largest interval on which the given function is increasing.d. R(x) = √ 2x - 1