Let $f\left(x\right)$ be a twice differentiable function in $\left[a,b\right]$, given that $f\left(x\right)$ and $f^{"}\left(x\right)$ has same sign in $\left[a,b\right]$.
Statement I: $f^{\text{'}}\left(x\right)=0$ has at the most one real root in $\left[a,b\right]$.
Statement II: An increasing function can intersect the $x$-axis at the most once.

For the following questions, choose the correct answers from the codes (a), (b), (c) and (d) defined as follows:
Let $f\left(0\right)=0,f\left(\frac{\pi }{2}\right)=1,f\left(\frac{3\pi }{2}\right)=-1$ be a continuous and twice differentiable function.
Statement I: $\left|f^{\text{'}}\left(x\right)\right|\le 1$ for atleast one $x\in \left(0,\frac{3\pi }{2}\right)$ because
Statement II: According to Rolle's theorem, if $y=g\left(x\right)$ is continuous and differentiable $\forall x\in \left[a,b\right]$ and $g\left(a\right)=g\left(b\right)$, then there exists atleast one $c$ such that $g^{\text{'}}\left(c\right)=0$.

For the given questions choose the correct answer from the given options defined as follows:

$f\left(x\right)$ is a polynomial of degree $3$ passing through origin having local extrema at $x=\pm 2$.

Statement I: Ratio of area in which $f\left(x\right)$ cuts the circle $x^2+y^2=36$ is $1:1$.

Statement II: Both $y=f\left(x\right)$ and the circle are symmetric about origin.