the function (f(x)=-2x^{3}+15x^{2}+84x - 3) is increasing on the interval ( , ), is decreasing on the…

Explanation:

Step1: Find the derivative of the function

The function is (f(x)=- 2x^{3}+15x^{2}+84x - 3). Using the power - rule ((x^n)^\prime=nx^{n - 1}), we get (f^\prime(x)=-6x^{2}+30x + 84).

Step2: Set the derivative equal to zero

Set (f^\prime(x)=0), so (-6x^{2}+30x + 84 = 0). Divide through by (-6) to simplify: (x^{2}-5x - 14=0).

Step3: Factor the quadratic equation

Factor (x^{2}-5x - 14) as ((x - 7)(x+2)=0). Then, by the zero - product property, (x = 7) or (x=-2).

Step4: Determine the intervals of increase and decrease

We consider the intervals ((-\infty,-2)), ((-2,7)) and ((7,\infty)). Test a value in the interval ((-\infty,-2)), say (x=-3). Then (f^\prime(-3)=-6\times(-3)^{2}+30\times(-3)+84=-54-90 + 84=-60<0), so the function is decreasing on ((-\infty,-2)). Test a value in the interval ((-2,7)), say (x = 0). Then (f^\prime(0)=-6\times0^{2}+30\times0 + 84=84>0), so the function is increasing on ((-2,7)). Test a value in the interval ((7,\infty)), say (x = 8). Then (f^\prime(8)=-6\times8^{2}+30\times8 + 84=-384+240 + 84=-60<0), so the function is decreasing on ((7,\infty)). Since the function changes from increasing to decreasing at (x = 7), the relative maximum occurs at (x = 7).

Answer:

The function is increasing on the interval ((-2,7)), decreasing on the intervals ((-\infty,-2)) and ((7,\infty)), and has a relative maximum at (x = 7).