Problem $4.$ Find the local and global extreme values of the functions, if any.

$($a$)f(x)=-x^2-3x+3$

$($h$)f(x)=\frac{x^3}{3x^2+1}$

$($b$)f(x)=x^4-8x^2+16$

$($i$)f(x)=\sqrt[3]{x^3+1}$

$($c$)f(x)=x^4-4x^3+4x^2$

$($d$)f(x)=x-6\sqrt[\phantom{2}]{x-1}$

$($j$)f(x)=\sqrt[\phantom{2}]{|x|}=\{\begin{array}{cc}\sqrt[\phantom{2}]{-x},& x<mn>0</mn>\\ \sqrt[\phantom{2}]{x},& x0\end{array}$

$($e$)f(x)=x\sqrt[\phantom{2}]{8-x^2}$

$($f$)f(x)=x^{\frac{1}{3}}(x^2-4)$

$($k$)f(x)=5x^{\frac{2}{3}}-2x$

$($g$)f(x)=x^{\frac{2}{3}}(x+5)$

$($l$)f(x)=|x^2-1|$

Problem $5.$ Find the values of constants $a,b,$ and $c$ so that the graph of $y=a{x}^3+b{x}^2+cx$ has a local maximum at $x=3,$ local minimum at $x=-1,$ and inflection point at $(1,11).$ A point $(c,f(c))$ is called an inflection point of the graph of $y=f(x)$ if $f^{\text{'}\text{'}}(c)=0$ or $f^{\text{'}\text{'}}(c)$ fails to exist.