Proof of Rolle's Theorem

Q. Lef $f,g$ and $h$ be differentiable functions. If $f\left(0\right)=1,$ $g\left(0\right)=2,$ $h\left(0\right)=3$ and the derivatives of their pairwise products at $x=0$ are $\left(fg{)}^{\prime }\right(0)=6,$ $\left(gh{)}^{\prime }\right(0)=4$ and $\left(hf{)}^{\prime }\right(0)=5,$ then the value of $\left(fgh{)}^{\prime }\right(0)$ is

Q. Let $y=f\left(x\right)\text{and}y=g\left(x\right)$ be two differentiable function in $[0,2]$ such that $f\left(0\right)=3,f\left(2\right)=5,g\left(0\right)=1\text{and}g\left(2\right)=2.$ If there exists atleast one $cϵ(0,2)$ such that $f^{\prime }\left(c\right)=k{g}^{\prime }\left(c\right)$, then $k$ must be

1. 2
2. 1
3. 3
4. $\frac{1}{2}$

Q. For all twice differentiable functions $f:R\to R$, with $f\left(0\right)=f\left(1\right)=f^{\prime }\left(0\right)=0$,

1. $f^{\prime \prime }\left(x\right)=0$, at every point $x\in (0,1)$
2. $f^{\prime \prime }\left(x\right)\ne 0$, at every point $x\in (0,1)$
3. $f^{\prime \prime }\left(x\right)=0$, for some $x\in (0,1)$
4. $f^{\prime \prime }\left(0\right)=0$

Q.

Let f(x) = x + 1; where $xϵ[0,\infty ]$. Choose the right option.

1. f(x) is discontinuous at but removable

2. None of these

3. f(x) is discontinuous at

4. f(x) is continuous at

Q. Let $y=f\left(x\right)\text{and}y=g\left(x\right)$ be two differentiable function in $[0,2]$ such that $f\left(0\right)=3,f\left(2\right)=5,g\left(0\right)=1\text{and}g\left(2\right)=2.$ If there exists atleast one $cϵ(0,2)$ such that $f^{\prime }\left(c\right)=k{g}^{\prime }\left(c\right)$, then $k$ must be

1. 2
2. 3
3. $\frac{1}{2}$
4. 1

Q. Let $y=f\left(x\right)\text{and}y=g\left(x\right)$ be two differentiable function in $[0,2]$ such that $f\left(0\right)=3,f\left(2\right)=5,g\left(0\right)=1\text{and}g\left(2\right)=2.$ If there exists atleast one $cϵ(0,2)$ such that $f^{\prime }\left(c\right)=k{g}^{\prime }\left(c\right)$, then $k$ must be

1. 2
2. 3
3. $\frac{1}{2}$
4. 1

Q. Let $y=f\left(x\right)\text{and}y=g\left(x\right)$ be two differentiable function in $[0,2]$ such that $f\left(0\right)=3,f\left(2\right)=5,g\left(0\right)=1\text{and}g\left(2\right)=2.$ If there exists atleast one $cϵ(0,2)$ such that $f^{\prime }\left(c\right)=k{g}^{\prime }\left(c\right)$, then $k$ must be

1. 2
2. 3
3. $\frac{1}{2}$
4. 1