Question

the polynomial function g is defined by g(x)=-2x³ + 2x² + 5x - 2. use the aleks graphing calculator to find all the points (x, g(x)) where there is a local minimum. round to the nearest hundredth. if there is more than one point, enter them using the \and\ button.

Explanation:

Step1: Find the derivative of \(g(x)\)

The function is \(g(x)=-2x^{3}+2x^{2}+5x - 2\). Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(g^\prime(x)=-6x^{2}+4x + 5\).

Step2: Find the critical points

Set \(g^\prime(x) = 0\), so \(-6x^{2}+4x + 5=0\). Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx + c = 0\) (here \(a=-6\), \(b = 4\), \(c = 5\)), we have \(x=\frac{-4\pm\sqrt{16-4\times(-6)\times5}}{2\times(-6)}=\frac{-4\pm\sqrt{16 + 120}}{-12}=\frac{-4\pm\sqrt{136}}{-12}=\frac{-4\pm2\sqrt{34}}{-12}=\frac{2\pm\sqrt{34}}{6}\).
\(x_1=\frac{2+\sqrt{34}}{6}\approx\frac{2 + 5.83}{6}\approx1.30\), \(x_2=\frac{2-\sqrt{34}}{6}\approx\frac{2-5.83}{6}\approx - 0.64\).

Step3: Use the second - derivative test

Find the second - derivative \(g^{\prime\prime}(x)=-12x + 4\).
For \(x=\frac{2+\sqrt{34}}{6}\), \(g^{\prime\prime}(\frac{2+\sqrt{34}}{6})=-12\times\frac{2+\sqrt{34}}{6}+4=-4 - 2\sqrt{34}+4=-2\sqrt{34}<0\) (local maximum).
For \(x=\frac{2-\sqrt{34}}{6}\), \(g^{\prime\prime}(\frac{2-\sqrt{34}}{6})=-12\times\frac{2-\sqrt{34}}{6}+4=-4 + 2\sqrt{34}+4=2\sqrt{34}>0\) (local minimum).

Step4: Calculate \(g(x)\) at \(x=\frac{2-\sqrt{34}}{6}\)

\(g(\frac{2-\sqrt{34}}{6})=-2\times(\frac{2-\sqrt{34}}{6})^{3}+2\times(\frac{2-\sqrt{34}}{6})^{2}+5\times(\frac{2-\sqrt{34}}{6})-2\).
First, \((\frac{2-\sqrt{34}}{6})^{2}=\frac{4-4\sqrt{34}+34}{36}=\frac{38-4\sqrt{34}}{36}=\frac{19 - 2\sqrt{34}}{18}\).
\((\frac{2-\sqrt{34}}{6})^{3}=(\frac{2-\sqrt{34}}{6})\times\frac{19 - 2\sqrt{34}}{18}=\frac{38-4\sqrt{34}-19\sqrt{34}+2\times34}{108}=\frac{38-23\sqrt{34}+68}{108}=\frac{106-23\sqrt{34}}{108}\).
\(g(\frac{2-\sqrt{34}}{6})=-2\times\frac{106-23\sqrt{34}}{108}+2\times\frac{19 - 2\sqrt{34}}{18}+\frac{10 - 5\sqrt{34}}{6}-2\)
\(=\frac{-106 + 23\sqrt{34}}{54}+\frac{19 - 2\sqrt{34}}{9}+\frac{10 - 5\sqrt{34}}{6}-2\)
\(=\frac{-106 + 23\sqrt{34}+114-12\sqrt{34}+90 - 45\sqrt{34}-108}{54}\)
\(=\frac{(-106 + 114+90 - 108)+(23\sqrt{34}-12\sqrt{34}-45\sqrt{34})}{54}\)
\(=\frac{-10 - 34\sqrt{34}}{54}\approx\frac{-10-34\times5.83}{54}\approx\frac{-10-198.22}{54}\approx - 3.85\).

Answer:

\((-0.64,-3.85)\)