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.\n\n$f(x)=1 - 3x+x^{2}$\n\ndescribe any relative extrema. select the correct choice below and, if necessary, fill in the answer boxes in within your choice.\n\na. the relative maximum 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.)\n\nb. 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.)\n\nc. the relative maximum point(s) is/are and there are no relative minimum points.\n(simplify your answer. type as ordered pair, using integers or fractions. use a comma to separate answers as needed.)\n\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.\n\n$f(x)=1 - 3x+x^{2}$\n\ndescribe any relative extrema. select the correct choice below and, if necessary, fill in the answer boxes in within your choice.\n\na. the relative maximum 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.)\n\nb. 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.)\n\nc. the relative maximum point(s) is/are and there are no relative minimum points.\n(simplify your answer. type as ordered pair, using integers or fractions. use a comma to separate answers as needed.)\n\nd. there are no relative minimum points and there are no relative maximum points.

Answer

Explanation:

Step1: Find the derivative

Given $f(x)=1 - 3x+x^{2}$, the derivative $f'(x)=\frac{d}{dx}(1 - 3x+x^{2})$. Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $f'(x)=-3 + 2x$.

Step2: Find the critical points

Set $f'(x)=0$. So, $-3 + 2x=0$. Solving for $x$, we get $2x=3$, then $x=\frac{3}{2}$.

Step3: Determine the second - derivative

Find the second - derivative $f''(x)=\frac{d}{dx}(-3 + 2x)=2$. Since $f''(\frac{3}{2}) = 2>0$, the function has a relative minimum at $x=\frac{3}{2}$.

Step4: Find the value of the relative minimum

Substitute $x = \frac{3}{2}$ into the original function $f(x)$: $f(\frac{3}{2})=1-3\times\frac{3}{2}+(\frac{3}{2})^{2}=1-\frac{9}{2}+\frac{9}{4}=\frac{4 - 18+9}{4}=-\frac{5}{4}$.

Step5: Determine intervals of increase and decrease

Choose a test point in the interval $(-\infty,\frac{3}{2})$, say $x = 0$. Then $f'(0)=-3<0$, so the function is decreasing on the interval $(-\infty,\frac{3}{2})$. Choose a test point in the interval $(\frac{3}{2},\infty)$, say $x = 2$. Then $f'(2)=1>0$, so the function is increasing on the interval $(\frac{3}{2},\infty)$.

Answer:

B. The relative minimum point(s) is $(\frac{3}{2},-\frac{5}{4})$ and there are no relative maximum point(s).