Question

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

Accepted Answer

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. Find the first derivative $$f'(x) to $$determine where the function is increasing or decreasing. Use the power rule: $$f'(x) = \frac{d}{dx}(-3x^2$ + 18x + 1) = -6x + 18. $$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. 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. Based on the sign of $$f'(x)$$, conclude the largest open intervals where $$f(x) is $$increasing (where $$f'(x) \u003e 0) $$and where it is decreasing (where $$f'(x) \u003c 0).$$

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

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Key Concepts

Domain of a Function

Increasing and Decreasing Intervals

Using the Derivative to Determine Monotonicity