Question 15, 3.1.34 HW Score: $54.67 \%, 13.67$ of 25 points Part 1 of 3 Points: 0 of 1 Save 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. \[ f(x)=4+x+x^{2} \] 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 $\square$ 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 $\square$ 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 $\square$ 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.
Solution
Step 1 ：The function given is \(f(x) = 4 + x + x^{2}\).
Step 2 ：First, we find the derivative of the function, \(f'(x) = 2x + 1\).
Step 3 ：The critical points of the function are the solutions to \(f'(x) = 0\). Solving this equation gives us \(x = -1/2\).
Step 4 ：We then find the second derivative of the function, \(f''(x) = 2\).
Step 5 ：Since \(f''(-1/2) > 0\), the point \(x = -1/2\) is a relative minimum.
Step 6 ：The function is increasing where its derivative is positive, which is when \(x > -1/2\), and decreasing where its derivative is negative, which is when \(x < -1/2\).
Step 7 ：Thus, the relative minimum point is \((-1/2, 15/4)\) and there are no relative maximum points. The function is increasing for \(x > -1/2\) and decreasing for \(x < -1/2\).
Step 8 ：Final Answer: \(\boxed{\text{The relative minimum point is } (-1/2, 15/4) \text{ and there are no relative maximum points. The function is increasing for } x > -1/2 \text{ and decreasing for } x < -1/2.}\)
