$g\left(x\right)=ax^2+bx+c$ such that $g\left(0\right)=1, g'\left(1\right)=g\left(1\right)$ and $g'\left(-1\right)=g\left(-1\right)$.
Find the values of $a, b$ and $c$.

Let $\theta$ be the angle between the line tangent to the function $f\left(x\right)=-x^3-\frac{1}{2}x^2+x$ at $x=\frac{1}{\sqrt{3}}$ and the $x$-axis. Find $\theta$ in degrees.

$\theta=$ Check

Find the equations of the two lines tangent to the funtion $f\left(x\right)=x^2+3x$ that pass through the point $\left(0,-4\right)$.

$y=$ Check and $y=$ Check

$A$ and $B$ are two distinct points on the graph of $f\left(x\right)=kx^2$ where $k\ne 0$.
Let the $x$-coordinate of $A$ be $a$ and the $x$-coordinate of $B$ be $b$.
The line tangent to point $A$ and the line tangent to point $B$ intersect at point $C$.
Find the coordinates of point $C$ in terms of $a$ and $b$.

If $f'\left(x\right)+f\left(x\right)=x^3+2x^2+5x+4$, find $f\left(x\right)$.

$f\left(x\right)=$ Check

$f\left(x\right)=ax^2+bx-2$ where $a$ and $b$ are constants.
Find the values of $a$ and $b$ if $f\left(f'\left(x\right)\right)=f'\left(f\left(x\right)\right)$