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

Verified step by step guidance

1. Identify the function given: \(f(x) = -(x - 2)^2 - 5\). This is a quadratic function in vertex form, where the squared term is \((x - 2)^2\) and it is multiplied by a negative sign.
2. Find the first derivative \(f'(x)\) to determine where the function is increasing or decreasing. Use the chain rule: \(f'(x) = -2(x - 2)\).
3. Set the derivative equal to zero to find critical points: \(-2(x - 2) = 0\). Solve for \(x\) to find the critical value where the function changes behavior.
4. Analyze the sign of \(f'(x)\) on intervals determined by the critical point. For values of \(x\) less than the critical point, check if \(f'(x)\) is positive or negative to determine if \(f\) is increasing or decreasing there. Repeat for values greater than the critical point.
5. Conclude the largest open intervals where \(f(x)\) is increasing and where it is decreasing based on the sign of \(f'(x)\) on those intervals.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like ƒ(x) = -(x - 2)^2 - 5, the domain is all real numbers since the function is defined everywhere on the real line.

Increasing and Decreasing Intervals

A function is increasing on an interval if its output values rise as x increases, and decreasing if its output values fall as x increases. Identifying these intervals involves analyzing the behavior of the function’s graph or its derivative to see where the slope is positive or negative.

Using the Derivative to Determine Monotonicity

The derivative of a function gives the slope of the tangent line at any point. If the derivative is positive over an interval, the function is increasing there; if negative, the function is decreasing. For ƒ(x) = -(x - 2)^2 - 5, finding and analyzing ƒ'(x) helps locate these intervals.