Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = -(x - 2)^2 - 5

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1. Identify the type of function: The given function \( f(x) = -(x - 2)^2 - 5 \) is a quadratic function in the form \( f(x) = a(x-h)^2 + k \), where \( a = -1 \), \( h = 2 \), and \( k = -5 \).
2. Determine the vertex: The vertex of the quadratic function \( f(x) = a(x-h)^2 + k \) is at the point \( (h, k) \). For this function, the vertex is \( (2, -5) \).
3. Analyze the direction of the parabola: Since \( a = -1 \) is negative, the parabola opens downwards. This means the function is increasing on the left of the vertex and decreasing on the right.
4. Identify the interval where the function is increasing: The function is increasing on the interval \((-\infty, 2)\).
5. Identify the interval where the function is decreasing: The function is decreasing on the interval \((2, \infty)\).

Key Concepts

Function Behavior

Understanding how a function behaves involves analyzing its increasing and decreasing intervals. A function is increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, it is decreasing if the function's value at the second point is less than at the first. This behavior is determined by the function's derivative.

Derivative

The derivative of a function provides information about its rate of change. If the derivative is positive over an interval, the function is increasing; if negative, it is decreasing. For the given function, finding the derivative and determining where it is positive or negative will help identify the intervals of increase and decrease.

Critical Points

Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find the intervals of increase and decrease, one must first locate these critical points. Analyzing the sign of the derivative around these points will reveal the behavior of the function in the intervals formed by these critical points.