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

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Identify the function given in the problem. Since the problem references Example 8, recall the specific function from that example or write down the function you need to analyze for decreasing intervals.

Find the first derivative of the function, denoted as \(f'(x)\), because the sign of the derivative tells us where the function is increasing or decreasing.

Set the derivative equal to zero and solve for \(x\) to find critical points: solve \(f'(x) = 0\). These points divide the domain into intervals where the function's behavior may change.

Determine the sign of \(f'(x)\) on each interval between the critical points by choosing test points. If \(f'(x) < 0\) on an interval, then the function is decreasing there.

Write the largest open intervals where \(f'(x) < 0\) as the intervals where the function is decreasing. Express these intervals in interval notation.

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Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. Identifying the domain is essential before analyzing behavior like increasing or decreasing intervals, as it restricts where the function can be evaluated.

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Increasing and Decreasing Functions

A function is decreasing on an interval if, as x increases, the function values decrease. Formally, f is decreasing on an interval if for any two points x1 < x2, f(x1) ≥ f(x2). Recognizing these intervals helps understand the function's behavior and graph shape.

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Graphs of Common Functions

Using the Derivative to Determine Monotonicity

The derivative of a function indicates its rate of change. If the derivative f'(x) is negative over an interval, the function is decreasing there. Finding where f'(x) < 0 helps identify the largest open intervals where the function decreases.

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