$(1$ point$)$

Find the intervals on which the function is concave up or down, the points of inflection, and the critical points, and determine whether each critical point corresponds to a local minimum or maximum $($or neither$).$ Let

$\backslash [$

f$($x$)=-(3$ x$+3\backslash$sin $($x$)),0\backslash$leq x $\backslash$leq $2\backslash$pi

$\backslash ]$

What are the critical point$($s$)=$

What does the Second Derivative Test tell about the first critical point:

What does the Second Derivative Test tell about the second critical point: $?$

What are the inflection Point$($s$)=$

On the interval to the left of the critical point, f is and $\backslash (\backslash$mathrm$\{$f$\}^\{\backslash$prime$\}\backslash )$ is $($Include all points where $\backslash (\backslash$mathrm$\{$f$\}^\{\backslash$prime$\}\backslash )$ has this sign in the interval.$)$

On the interval to the right of the critical point, f is and $\backslash (\backslash$mathrm$\{$f$\}^\{\backslash$prime$\}\backslash )$ is $.($Include all points where $\backslash (\backslash$mathrm$\{$f$\}^\{\backslash$prime$\}\backslash )$ has this sign in the interval.$)$

On the interval to the left of the inflection point $\backslash ($ f $\backslash )$ is $.($Include only points where f has this concavity in the interval.$)$ On the interval $\backslash ((\backslash$pi$,2\backslash$pi$)\backslash )$ to the right of the inflection point f is $.($Include only points where f has this concavity in the interval.$)$