Use a graphing utility to graph the function over the indicated interval and approximate any local maxima and local
minima. Determine where the function is increasing and where it is decreasing. If necessary, round answers to two
decimal places.
-$f(x)=x^{3}-3 x+3,(-2,2)$
A)
local maximum at $ (1,1) $
local minimum at $ (-1,5) $
increasing on $ (-2,-1) $
decreasing on $ (-1,1) $

B) local maximum at $ (-1,5) $
local minimum at $ (1,1) $
increasing on $ (-2,-1) $ and $ (1,2) $
decreasing on $ (-1,1) $

C) local maximum at $ (1,1) $
local minimum at $ (-1,5) $
increasing on $ (-2,-1) $ and $ (1,2) $
decreasing on $ (-1,1) $

D)
$\begin{array}{l}
\text { local maximum at }(-1,5) \
\text { local minimum at }(1,1) \
\text { increasing on }(-1,1) \
\text { decreasing on }(-2,-1) \text { and }(1,2)
\end{array}$

Question

Use a graphing utility to graph the function over the indicated interval and approximate any local maxima and local
minima. Determine where the function is increasing and where it is decreasing. If necessary, round answers to two
decimal places.
-$f(x)=x^{3}-3 x+3,(-2,2)$
A)
local maximum at $ (1,1) $
 local minimum at $ (-1,5) $ 
increasing on $ (-2,-1) $
 decreasing on $ (-1,1) $

B) local maximum at $ (-1,5) $
local minimum at $ (1,1) $
increasing on $ (-2,-1) $ and $ (1,2) $
decreasing on $ (-1,1) $

C) local maximum at $ (1,1) $
local minimum at $ (-1,5) $
increasing on $ (-2,-1) $ and $ (1,2) $
decreasing on $ (-1,1) $

D) 
$\begin{array}{l}
\text { local maximum at }(-1,5) \
\text { local minimum at }(1,1) \
\text { increasing on }(-1,1) \
\text { decreasing on }(-2,-1) \text { and }(1,2)
\end{array}$

Quizplus · Accepted Answer

To find local maxima and minima, we first find the derivative of $f(x) = x^3 - 3x + 3$, which is $f'(x) = 3x^2 - 3$. Setting $f'(x) = 0$ gives $x^2 = 1$, so $x = \pm 1$. Testing these values in the second derivative, $f''(x) = 6x$, we find that $f''(-1) = -6 < 0$ (indicating a local maximum at $x = -1$) and $f''(1) = 6 > 0$ (indicating a local minimum at $x = 1$). Plugging $x = -1$ and $x = 1$ into the original function gives the points $(-1, 5)$ and $(1, 1)$, respectively. To determine where the function is increasing or decreasing, we consider the sign of $f'(x)$ in the intervals divided by $x = -1$ and $x = 1$. The function is increasing on $(-2, -1)$ and $(1, 2)$, and decreasing on $(-1, 1)$.

Use a Graphing Utility to Graph the Function Over the Indicated