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)=22 + 12x-2x^{2}) \ndescribe any relative extrema. select the correct choice below and if necessary, fill in the answer box(es) to within your choice. \na. the relative minimum point(s) is/are and the relative maximum point(s) is/are \n(simplify your answers. type ordered - pairs, using integers or fractions. use a comma to separate answers as needed.) \nb. the relative maximum point(s) is/are and there are no relative minimum points. \n(simplify your answer. type an ordered - pair, using integers or fractions. use a comma to separate answers as needed.) \nc. the relative minimum point(s) is/are and there are no relative maximum points. \n(simplify your answer. type an ordered - pair, using integers or fractions. use a comma to separate answers as needed.) \nd. 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)=22 + 12x-2x^{2}) \ndescribe any relative extrema. select the correct choice below and if necessary, fill in the answer box(es) to within your choice. \na. the relative minimum point(s) is/are and the relative maximum point(s) is/are \n(simplify your answers. type ordered - pairs, using integers or fractions. use a comma to separate answers as needed.) \nb. the relative maximum point(s) is/are and there are no relative minimum points. \n(simplify your answer. type an ordered - pair, using integers or fractions. use a comma to separate answers as needed.) \nc. the relative minimum point(s) is/are and there are no relative maximum points. \n(simplify your answer. type an ordered - pair, using integers or fractions. use a comma to separate answers as needed.) \nd. there are no relative minimum points and there are no relative maximum points.

Answer

Explanation:

Step1: Find the derivative

The function is $g(x)=22 + 12x-2x^{2}$. Using the power - rule for differentiation $\frac{d}{dx}(x^{n})=nx^{n - 1}$, the derivative $g'(x)=\frac{d}{dx}(22)+\frac{d}{dx}(12x)-\frac{d}{dx}(2x^{2})=0 + 12-4x=12 - 4x$.

Step2: Find the critical points

Set $g'(x)=0$. So, $12 - 4x = 0$. Solving for $x$: [ \begin{align*} 12-4x&=0\ -4x&=-12\ x& = 3 \end{align*} ]

Step3: Determine the second - derivative

Differentiate $g'(x)=12 - 4x$ with respect to $x$. The second - derivative $g''(x)=\frac{d}{dx}(12 - 4x)=-4$. Since $g''(3)=-4<0$, the function has a relative maximum at $x = 3$.

Step4: Find the value of the relative maximum

Substitute $x = 3$ into the original function $g(x)=22 + 12x-2x^{2}$. Then $g(3)=22+12\times3-2\times3^{2}=22 + 36-18=40$. So the relative maximum point is $(3,40)$ and there are no relative minimum points.

Step5: Determine the intervals of increase and decrease

Choose a test point in the interval $(-\infty,3)$, say $x = 0$. Then $g'(0)=12-4\times0 = 12>0$, so the function is increasing on the interval $(-\infty,3)$. Choose a test point in the interval $(3,\infty)$, say $x = 4$. Then $g'(4)=12-4\times4=12 - 16=-4<0$, so the function is decreasing on the interval $(3,\infty)$.

Answer:

B. The relative maximum point(s) is (3,40) and there are no relative minimum points.