Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = -3x² + 18x + 1

Verified step by step guidance

1. Identify the function given: \(f(x) = -3x^2 + 18x + 1\). Since this is a quadratic function, it is a parabola opening downward because the coefficient of \(x^2\) is negative.
2. Find the first derivative \(f'(x)\) to determine where the function is increasing or decreasing. Use the power rule: \(f'(x) = \frac{d}{dx}(-3x^2 + 18x + 1) = -6x + 18\).
3. Set the derivative equal to zero to find critical points: \(-6x + 18 = 0\). Solve for \(x\) to find the critical point(s) where the function could change from increasing to decreasing or vice versa.
4. Determine the sign of \(f'(x)\) on intervals divided by the critical point. For values of \(x\) less than the critical point, plug a test value into \(f'(x)\) to check if it is positive (increasing) or negative (decreasing). Repeat for values greater than the critical point.
5. Based on the sign of \(f'(x)\), conclude the largest open intervals where \(f(x)\) is increasing (where \(f'(x) > 0\)) and where it is decreasing (where \(f'(x) < 0\)).

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) = -3x² + 18x + 1, 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 over an interval, the function is increasing there; if negative, the function is decreasing. For ƒ(x) = -3x² + 18x + 1, finding ƒ'(x) and solving inequalities helps find these intervals.