Use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 2 x + 3$
A)relative maximum: (2.58, 0.39)relative minimum: (7.27, -1.72)
B)relative maximum: (0.39, 2.58)relative minimum: (-1.72, 7.27)
C)relative maximum: (-1.72, 7.27)relative minimum: (0.39, 2.58)
D)relative maximum: (7.27, -1.72)relative minimum: (2.58, 0.39)
E)relative maximum: (2.58, 28.33)relative minimum: (7.27, 478.41)

Question

Quizplus · Accepted Answer

To find the relative maximum and minimum of the function $f(x) = x^3 + 2x^2 - 2x + 3$, we first find its derivative $f'(x) = 3x^2 + 4x - 2$ and solve $f'(x) = 0$ to find critical points. Solving this gives $x \approx -1.72$ and $x \approx 0.39$. Using the second derivative test or analyzing the sign changes of $f'(x)$ around these points, we can determine that $x \approx -1.72$ corresponds to a relative maximum and $x \approx 0.39$ to a relative minimum. Substituting these $x$ values back into $f(x)$ gives the respective $y$ values, leading to the correct answer.

Use a Graphing Utility to Graph the Function and Approximate f(x)=x3+2x2−2x+3f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 2 x + 3f(x)=x3+2x2−2x+3

Use a Graphing Utility to Graph the Function and Approximate $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 2 x + 3$