For all real values of $a_{0}, a_{1}, a_{2}, a_{3}$ satisfying $a_{0}+\frac{a_{1}}{2}+\frac{a_{2}}{3}+\frac{a_{3}}{4}=0$, the equation $a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}=0$ has a real root in the interval

Given that $f(x)$ is continuously differentiable on $a \le x \le b$ where $a < b, f(a) < 0$ and $f(b) > 0$,which of the following are always true?
$(i)$ $f(x)$ is bounded on $a \le x \le b$.
$(ii)$ The equation $f(x) = 0$ has at least one solution in $a < x < b$.
$(iii)$ The maximum and minimum values of $f(x)$ on $a \le x \le b$ occur at points where $f'(c) = 0$.
$(iv)$ There is at least one point $c$ with $a < c < b$ where $f'(c) > 0$.
$(v)$ There is at least one point $d$ with $a < d < b$ where $f'(d) < 0$.

If $f(x)$ is a twice differentiable polynomial function such that $f(1) = 1, f(2) = 4, f(3) = 9$,then:

Consider the function $f(x)=2x^3-3x^2-x+1$ and the intervals $I_1=[-1,0]$, $I_2=[0,1]$, $I_3=[1,2]$, $I_4=[-2,-1]$. Then,

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function such that $f(0)=f(1)=f^{\prime}(0)=0$. Then:

If $f(x) = \log(\sin x)$,$x \in \left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$,then the value of $c$ by applying Lagrange's Mean Value Theorem $(LMVT)$ is: