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Q$1)(30$ pts$.)$ Let $f$:$\frac{R}{\frac{?}{?}}\{-3,5\}longrightarrowR$ be a function satisfying the following conditions:

$f(0)=1,f(7)=0,$

$f^{\text{'}}(x)<mn>0</mn>$ for

$\underset{x-3}{lim}f(x)=2$

$f^{\text{'}\text{'}}(7)=0$

$(a)$ Find the critical number$(s)$ and the intervals where $fis$ increasing and the intervals where $fis$ decreasing. Determine the local maximum and minimum values $off,if$ there exist any.

The critical number$(s)offi\frac{s}{a}re$

$fis$ decreasing $on$ the interva

$fis$ increasing $on$ the interval

The local min value$(s)offi\frac{s}{a}req,$ and

The local max value$(s)offi\frac{s}{a}re(write$ none $if$ they $do$ not exist$).-3,$ and $5$

$\underset{x-3}{lim}f(x)=2$

$f^{\text{'}\text{'}}(7)=0$

$(a)$ Find the critical number$(s)$ and the intervals where $fis$ increasing and the intervals where $fis$ decreasing. Determine the local maximum and minimum values $off,if$ there exist any.

The critical number$(s)offi\frac{s}{a}re$

$fis$ decreasing $on$ the interva

$fis$ increasing $on$ the interval

The local min value$(s)offi\frac{s}{a}req,$ and

The local max value$(s)offi\frac{s}{a}re(write$ none $if$ they $do$ not exist$)\mathrm{.}0$

$f^{\text{'}}(0)=0$

$\underset{x{5}^{-}}{lim}f(x)=,\underset{x{5}^{+}}{lim}f(x)=$

$f^{\text{'}\text{'}}(x)<mn>0</mn>$ for and $x>7$

$\underset{x}{lim}f(x)=-,\underset{x-}{lim}f(x)=4$

$f^{\text{'}\text{'}}(x)>0$ for $-3,$ and $5$

$\underset{x-3}{lim}f(x)=2$

$f^{\text{'}\text{'}}(7)=0$

$(a)$ Find the critical number$(s)$ and the intervals where $fis$ increasing and the intervals where $fis$ decreasing. Determine the local maximum and minimum values $off,if$ there exist any.

The critical number$(s)offi\frac{s}{a}re$

$fis$ decreasing $on$ the interva

$fis$ increasing $on$ the interval

The local min value$(s)offi\frac{s}{a}req,$ and

The local max value$(s)offi\frac{s}{a}re(write$ none $if$ they $do$ not exist and $x>5$

$f(-3)$ and $f(5)$ are undefined

$f^{\text{'}}(x)>0$ for $0$

$f^{\text{'}}(0)=0$

$\underset{x{5}^{-}}{lim}f(x)=,\underset{x{5}^{+}}{lim}f(x)=$

$f^{\text{'}\text{'}}(x)<mn>0</mn>$ for and $x>7$

$\underset{x}{lim}f(x)=-,\underset{x-}{lim}f(x)=4$

$f^{\text{'}\text{'}}(x)>0$ for $-3,$ and $5$

$\underset{x-3}{lim}f(x)=2$

$f^{\text{'}\text{'}}(7)=0$

$(a)$ Find the critical number$(s)$ and the intervals where $fis$ increasing and the intervals where $fis$ decreasing. Determine the local maximum and minimum values $off,if$ there exist any.

The critical number$(s)offi\frac{s}{a}re$

$fis$ decreasing $on$ the interva

$fis$ increasing $on$ the interval

The local min value$(s)offi\frac{s}{a}req,$ and

The local max value$(s)offi\frac{s}{a}re(write$ none $if$ they $do$ not exist$).$