the polynomial function f is defined by f(x)=x³ + 2x² - 2x - 3. use the aleks graphing calculator to find…

Explanation:

Step1: Find the derivative

The derivative of $f(x)=x^{3}+2x^{2}-2x - 3$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=3x^{2}+4x - 2$.

Step2: Set the derivative equal to zero

We solve the quadratic equation $3x^{2}+4x - 2 = 0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here, $a = 3$, $b = 4$, and $c=-2$. So $x=\frac{-4\pm\sqrt{4^{2}-4\times3\times(-2)}}{2\times3}=\frac{-4\pm\sqrt{16 + 24}}{6}=\frac{-4\pm\sqrt{40}}{6}=\frac{-4\pm2\sqrt{10}}{6}=\frac{-2\pm\sqrt{10}}{3}$.

Step3: Determine local minimum

We use the second - derivative test. The second - derivative $f''(x)=6x + 4$. For $x_1=\frac{-2+\sqrt{10}}{3}\approx\frac{-2 + 3.162}{3}=\frac{1.162}{3}\approx0.39$, $f''(0.39)=6\times0.39+4=2.34 + 4=6.34>0$, so there is a local minimum at $x=\frac{-2+\sqrt{10}}{3}\approx0.39$. $f(0.39)=(0.39)^{3}+2\times(0.39)^{2}-2\times0.39 - 3=0.059+0.304 - 0.78 - 3=-3.417\approx - 3.42$.

Answer:

$(0.39, - 3.42)$