the polynomial function g is defined by g(x)=-3x^3 + 3x^2+7x - 3. use the aleks graphing calculator to find…

Explanation:

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

The derivative (g'(x)) of (g(x)=- 3x^{3}+3x^{2}+7x - 3) is (g'(x)=-9x^{2}+6x + 7) using the power - rule ((x^n)'=nx^{n - 1}).

Step2: Set the derivative equal to zero

We solve the quadratic equation (-9x^{2}+6x + 7 = 0). The quadratic formula for (ax^{2}+bx + c = 0) is (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}). Here, (a=-9), (b = 6), and (c = 7). So (x=\frac{-6\pm\sqrt{6^{2}-4\times(-9)\times7}}{2\times(-9)}=\frac{-6\pm\sqrt{36 + 252}}{-18}=\frac{-6\pm\sqrt{288}}{-18}=\frac{-6\pm12\sqrt{2}}{-18}=\frac{1\pm\sqrt{2}}{3}).

Step3: Determine which value gives a local maximum

We use the second - derivative test. The second - derivative (g''(x)=-18x + 6). For (x_1=\frac{1+\sqrt{2}}{3}\approx\frac{1 + 1.414}{3}=\frac{2.414}{3}\approx0.80), (g''(x_1)=-18\times\frac{1+\sqrt{2}}{3}+6=-6(1 + \sqrt{2})+6=-6 - 6\sqrt{2}+6=-6\sqrt{2}<0), so (x=\frac{1+\sqrt{2}}{3}) gives a local maximum. For (x_2=\frac{1-\sqrt{2}}{3}\approx\frac{1-1.414}{3}=\frac{-0.414}{3}\approx - 0.14), (g''(x_2)=-18\times\frac{1-\sqrt{2}}{3}+6=-6(1-\sqrt{2})+6=-6 + 6\sqrt{2}+6=6\sqrt{2}>0), so (x=\frac{1-\sqrt{2}}{3}) gives a local minimum. When (x=\frac{1+\sqrt{2}}{3}\approx0.80), (g(x)=-3\times(\frac{1 + \sqrt{2}}{3})^{3}+3\times(\frac{1+\sqrt{2}}{3})^{2}+7\times\frac{1+\sqrt{2}}{3}-3). ((\frac{1+\sqrt{2}}{3})^{2}=\frac{1 + 2\sqrt{2}+2}{9}=\frac{3 + 2\sqrt{2}}{9}), ((\frac{1+\sqrt{2}}{3})^{3}=(\frac{1+\sqrt{2}}{3})\times(\frac{3 + 2\sqrt{2}}{9})=\frac{3+2\sqrt{2}+3\sqrt{2}+4}{27}=\frac{7 + 5\sqrt{2}}{27}). (g(x)=-3\times\frac{7 + 5\sqrt{2}}{27}+3\times\frac{3 + 2\sqrt{2}}{9}+\frac{7(1+\sqrt{2})}{3}-3) (=-\frac{7 + 5\sqrt{2}}{9}+\frac{3 + 2\sqrt{2}}{3}+\frac{7 + 7\sqrt{2}}{3}-3) (=-\frac{7 + 5\sqrt{2}}{9}+\frac{9 + 6\sqrt{2}+21 + 21\sqrt{2}}{9}-3) (=-\frac{7 + 5\sqrt{2}}{9}+\frac{30+27\sqrt{2}}{9}-3) (=\frac{-7 - 5\sqrt{2}+30 + 27\sqrt{2}}{9}-3) (=\frac{23 + 22\sqrt{2}}{9}-3=\frac{23 + 22\sqrt{2}-27}{9}=\frac{22\sqrt{2}-4}{9}\approx\frac{22\times1.414-4}{9}=\frac{31.108 - 4}{9}=\frac{27.108}{9}\approx3.01)

Answer:

((x,g(x))=(0.80,3.01))