find any relative extrema of the function. list each extremum along with the x - value at which it occurs…

find any relative extrema of the function. list each extremum along with the x - value at which it occurs. identify intervals over which the function is increasing and over which it is decreasing. then sketch a graph of the function. g(x)=-x³ + x² + x - 1 describe any relative extrema. select the correct choice below and, if necessary, fill in the answer box(es) to within your choice. a. the relative minimum point(s) is/are and there are no relative maximum points. (simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed.) b. the relative minimum point(s) is/are and the relative maximum point(s) is/are (simplify your answers. type ordered pairs, using integers or fractions. use a comma to separate answers as needed.) c. the relative maximum point(s) is/are and there are no relative minimum points. (simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed.) d. there are no relative minimum points and there are no relative maximum points.

find any relative extrema of the function. list each extremum along with the x - value at which it occurs. identify intervals over which the function is increasing and over which it is decreasing. then sketch a graph of the function. g(x)=-x³ + x² + x - 1 describe any relative extrema. select the correct choice below and, if necessary, fill in the answer box(es) to within your choice. a. the relative minimum point(s) is/are and there are no relative maximum points. (simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed.) b. the relative minimum point(s) is/are and the relative maximum point(s) is/are (simplify your answers. type ordered pairs, using integers or fractions. use a comma to separate answers as needed.) c. the relative maximum point(s) is/are and there are no relative minimum points. (simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed.) d. there are no relative minimum points and there are no relative maximum points.

Answer

Explanation:

Step1: Find the derivative

Differentiate $G(x)=-x^{3}+x^{2}+x - 1$ using the power - rule. $G'(x)=-3x^{2}+2x + 1$.

Step2: Set the derivative equal to zero

Solve $-3x^{2}+2x + 1 = 0$. Factor the quadratic equation: $-3x^{2}+3x - x+1 = 0$, $-3x(x - 1)-(x - 1)=0$, $(-3x - 1)(x - 1)=0$. So $x=-\frac{1}{3}$ or $x = 1$.

Step3: Use the second - derivative test

Differentiate $G'(x)$ to get the second - derivative $G''(x)=-6x + 2$. When $x=-\frac{1}{3}$, $G''(-\frac{1}{3})=-6\times(-\frac{1}{3})+2=4>0$. So $G(x)$ has a relative minimum at $x = -\frac{1}{3}$. $G(-\frac{1}{3})=-(-\frac{1}{3})^{3}+(-\frac{1}{3})^{2}+(-\frac{1}{3})-1=-\frac{32}{27}$. When $x = 1$, $G''(1)=-6\times1+2=-4<0$. So $G(x)$ has a relative maximum at $x = 1$. $G(1)=-1 + 1+1 - 1=0$.

Step4: Determine increasing and decreasing intervals

Test intervals using the first - derivative. The intervals are $(-\infty,-\frac{1}{3})$, $(-\frac{1}{3},1)$ and $(1,\infty)$. For $x<- \frac{1}{3}$, let $x=-1$, $G'(-1)=-3-2 + 1=-4<0$, so $G(x)$ is decreasing on $(-\infty,-\frac{1}{3})$. For $-\frac{1}{3}<x<1$, let $x = 0$, $G'(0)=1>0$, so $G(x)$ is increasing on $(-\frac{1}{3},1)$. For $x>1$, let $x = 2$, $G'(2)=-3\times4 + 2\times2+1=-7<0$, so $G(x)$ is decreasing on $(1,\infty)$.

Answer:

B. The relative minimum point(s) is/are $(-\frac{1}{3},-\frac{32}{27})$ and the relative maximum point(s) is/are $(1,0)$