#### Answer

$a.$ The value of the derivative is $f'(0)=0$.
$b.$ The equation is $y=0$ (the $x$ axis).
$c.$ The graph is on the figure below.

#### Work Step by Step

$a.$ Using the definition of the derivative with $a=0$ we have
$$f'(0)=\lim_{h\to0}\frac{f(0+h)-f(0)}{h}=\lim_{h\to0}\frac{3(0+h)^2-3\cdot0}{h}=\lim_{h\to0}\frac{3h^2}{h}=\lim_{h\to0}3h=3\cdot0=0.$$
$b.$ The equation of the tangent line through the point $(a,f(a))$ is given by $y-f(a)=f'(a)(x-a)$. Using $a=0$, $f(a) =f(1)= 3\cdot0^2=0$ and $f'(a)=f'(1)=0$ we have
$$y-0=0\cdot(x-0)\Rightarrow y=0.$$
$c.$ The graph is on the figure below. The function is graphed by the solid line and the tangent is dashed (the tangent is $x$ axis actually).