Mean Value Theorems

$A\left(0,1\right),B\left(\frac{\pi }{2},1\right)$ are two points on the graph given by $y=2 \sin x+\cos 2x$. If there exists a point $P$ on the curve between $A \& B$ such that tangent at $P$ is parallel to $AB$. Having co-ordinates $\left(\frac{\pi }{a}, \frac{3}{b}\right),$ find $\left(a+b\right)$.

Which of the following inequalities are valid ?

For the function $f\left(x\right)=x {\cot }^{-1}x, x\ge 0$

Which of the following is correct?

What can be said about the quadratic equation $a{x}^2+bx+c=0$ if $2a+3b+6c=0,$ $\left(a\mathrm{,} b\mathrm{,} c\in R\right) ?$

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Consider the polynomial function
$f\left(x\right)=\frac{x^7}{7}-\frac{x^6}{6}+\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x$
Statement I: The equation $f\left(x\right)=0$ cannot have two or more roots.
Statement II: Rolle's theorem is not applicable for $y=f\left(x\right)$ on any interval
$\left[a, b\right]$, where $a, b\in R$.

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Statement I: If $g\left(x\right)$ is a differentiable function $g\left(1\right)\ne 0, g\left(-1\right)\ne 0$ and Rolle's theorem is not applicable $f\left(x\right)=\frac{x^2-1}{g\left(x\right)}$ in $\left[-\mathrm{1,} 1\right]$ then $g\left(x\right)$ has atleast one root in $\left(-\mathrm{1,} 1\right)$.
Statement II: If $f\left(a\right)=f\left(b\right)$, then Rolle's theorem is applicable for $x\in \left(a, b\right).$

For every twice differentiable function $f:\mathrm{ℝ}\to \left[-2, 2\right]$ with ${\left(f\left(0\right)\right)}^2+{\left(f^{\prime }\left(0\right)\right)}^2=85$, which of the following statement(s) is (are) TRUE?