Thinking $(16$ marks$)$

$6.$ Determine the critical points of the given function and classify the nature of each critical point.

$($i$.$e$.$ is the critical point a local $mi\frac{n}{?}$ local $ma\frac{x}{?}$ or stationary point$)$

a$)$

b$)$

c$)$

d$)$

e$)$

f$)$

$-4x^3-12x^2=0$

$-4x^2(x+3)=0$

$3,x=0orx=-3$

$x=0f^{\text{'}\text{'}}(x)=-12x^2-24x$

$f^{\text{'}\text{'}}(0)=-12(0{)}^2-24(0)=0$

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

:$.x=0is$ the

local maximum

$7.$ Find the interval$($s$)$ over which the function $f(x)=3x^5-5x^3$ is concave down.

$\frac{?}{\text{b}\text{a}\text{r}}(3)$