determine the critical values of (f(x)=8x^{3}+57x^{2}-30x + 5). separate your answers with commas. leave…

Explanation:

Step1: Find the derivative of f(x)

The derivative of (f(x)=8x^{3}+57x^{2}-30x + 5) using the power - rule ((x^n)^\prime=nx^{n - 1}) is (f^\prime(x)=24x^{2}+114x-30).

Step2: Set the derivative equal to zero

Set (f^\prime(x)=0), so (24x^{2}+114x - 30=0). Divide through by (6) to simplify: (4x^{2}+19x - 5=0).

Step3: Solve the quadratic equation

For a quadratic equation (ax^{2}+bx + c = 0) ((a = 4), (b=19), (c=-5)), use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}). First, calculate the discriminant (\Delta=b^{2}-4ac=(19)^{2}-4\times4\times(-5)=361 + 80=441). Then (x=\frac{-19\pm\sqrt{441}}{2\times4}=\frac{-19\pm21}{8}).

Step4: Find the two solutions

For the plus - sign: (x=\frac{-19 + 21}{8}=\frac{2}{8}=\frac{1}{4}). For the minus - sign: (x=\frac{-19-21}{8}=\frac{-40}{8}=-5).

Answer:

(-5,\frac{1}{4})