Determine the largest open intervals of the domain over which each function is (a) increasing See Example 8.

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Identify the given function for which you need to determine the intervals of increase. The problem refers to 'each function,' so start by clearly writing down the function(s) involved.

Recall that a function is increasing on intervals where its first derivative is positive. Therefore, find the first derivative of the function, denoted as \(f'(x)\).

Set up the inequality \(f'(x) > 0\) to find where the function is increasing. Solve this inequality to determine the values of \(x\) for which the derivative is positive.

Analyze the critical points where \(f'(x) = 0\) or where \(f'(x)\) is undefined, as these points can mark the boundaries of intervals where the function changes from increasing to decreasing or vice versa.

Combine the results to write the largest open intervals on the domain where \(f'(x) > 0\), which correspond to the intervals where the original function is increasing.

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Domain of a Function

The domain of a function is the set of all input values (usually x-values) for which the function is defined. Identifying the domain is essential before analyzing behavior like increasing or decreasing intervals, especially for trigonometric functions that may have restricted domains due to their definitions or transformations.

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Increasing and Decreasing Functions

A function is increasing on an interval if, as the input increases, the output also increases. Formally, f is increasing on an interval if for any x1 < x2 in that interval, f(x1) < f(x2). Understanding this concept helps in determining where the function rises or falls.

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Use of Derivatives to Determine Monotonicity

The derivative of a function indicates its rate of change. If the derivative is positive over an interval, the function is increasing there; if negative, it is decreasing. Calculating and analyzing the derivative is a key method to find the largest intervals where the function is increasing.

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