Question

determine the critical values of (f(x)=8x^{3}+57x^{2}-30x + 5). separate your answers with commas. leave answers as integers or reduced fractions. the critical values are:

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:

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