Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = -2x² - 8x - 7

Verified step by step guidance

1. Identify the function given: \(f(x) = -2x^2 - 8x - 7\). Since this is a quadratic function, its graph is a parabola.
2. Find the first derivative \(f'(x)\) to determine where the function is increasing or decreasing. Use the power rule: \(f'(x) = \frac{d}{dx}(-2x^2) + \frac{d}{dx}(-8x) + \frac{d}{dx}(-7)\).
3. Simplify the derivative: \(f'(x) = -4x - 8\). This derivative tells us the slope of the tangent line at any point \(x\).
4. Set the derivative equal to zero to find critical points: \(-4x - 8 = 0\). Solve for \(x\) to find where the slope changes from positive to negative or vice versa.
5. Use the critical point to divide the domain into intervals. Test values from each interval in \(f'(x)\) to determine if the function is increasing (where \(f'(x) > 0\)) or decreasing (where \(f'(x) < 0\)) 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) = -2x² - 8x - 7, the domain is all real numbers since polynomials are 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 slope or derivative over different parts of the domain.

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 on an interval, the function is increasing there; if negative, the function is decreasing. For ƒ(x) = -2x² - 8x - 7, finding ƒ'(x) and solving inequalities helps find these intervals.