Mean Value Theorem

The point on the curve $y=x^2$, where the tangent is parallel to the line joining the points $(1, 1)$ and $(2, 4)$ is

The point on the curve $y=x^3-3x$, where the tangent to the curve is parallel to the chord joining $(1,-2)$ and $(2, 2)$ is 

Find a point on the curve $y=x^3$, where the tangent to the curve is parallel to the chord joining the points $(1, 1)$ and $(3, 27).$

Let $f\left(x\right)=\sin x+\left(x^3-3x^2+4x-2\right)\cos x$ for $x\in (0,1).$ Consider the following statements
I. $f$ has a zero in $\left(0, 1\right)$
II. $f$ is monotone in $\left(0, 1\right)$
Then

Consider the following: 

1. The arithemetic mean of two unequal postive numbers is always greater than their geometric mean. 

2. The geometric mean of two unequal positive numbers is alwaya greater than their harmonic mean. 

Which of the above statements is/are correct? 

What are the values if c for which Rolle's theorem for the function $f\left(x\right)=x^3-3x^2+2x$ in the interval [0, 2] is verified?

Let $f\left(x\right)=\{\begin{array}{cc}x& ,0\le x<1\\ 2-x& ,1\le x\le 2 \end{array}$$,$ then, Rolle's theorem is not applicable to $f\left(x\right)$ in $\left(0, 2\right)$ because

If the function $f\left(x\right)=x^3-6x^2+ax+b$ satisfies Rolle's theorem in the interval

$[1, 3]$ and $f^{\prime }\left(\frac{2\sqrt{3}+1}{\sqrt{3}}\right)=0$, then:

Let $f\left(x\right)=x^3$ use mean value theorem to write $\frac{f\left(x+h\right)-f\left(x\right)}{h}=f^{\text{'}}\left(x+\theta h\right)$ with $0<\theta <1$. If $x\ne 0$, then $\underset{h\to 0}{\lim }\theta$ is equal to:

If $f\left(x\right)=x^2-2x+4$ on $[1, 5]$, then the value of a constant $c$ such that $\frac{f\left(5\right)-f\left(1\right)}{5-1}=f^{\prime }\left(c\right)$, is

Let $f\left(x\right)=a_5{x}^5+a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x$, where $a_i$ are real and $f\left(x\right) = 0$ has a positive root ${\alpha }_0$. Then

Rolle's theorem is not applicable for the function $f\left(x\right)=\left|x\right|$ in the interval $\left[-1, 1\right]$ beacuse

If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_{1}x=0$, $a_1\ne 0, n\ge 2$, has a positive root $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+\left(n-1\right)a_{n-1}{x}^{n-2}+...+a_1=0$  has a positive root, which is