Intermediate Value of Theorem

Q. If $f\left(x\right)$ is continuous and differentiable over $[-2,5]$ and $-4\le f^{\prime }\left(x\right)\le 3$ for all $x$ in $(-2,5),$ then the greatest possible value of $f\left(5\right)-f(-2)$ is

1. $7$
2. $9$
3. $21$
4. $15$

Q. Let $f\left(x\right)$ and $g\left(x\right)$ be differentiable for $0\le x\le 1$, such that $f\left(0\right)=0,g\left(0\right)=0,f\left(1\right)=6$. Let there exists a real number $c$ in $(0,1)$ such that $f^{\prime }\left(c\right)=2g^{\prime }\left(c\right),$ then the value of $g\left(1\right)$ must be

1. $1$
2. $3$
3. $-2$
4. $-1$

Q. Let $f\left(x\right)$ be continous and differentiable function for all real values.
$f(x+y)=f\left(x\right)-3xy+f\left(y\right)$. If $\underset{h\to 0}{lim}\frac{f\left(h\right)}{h}=7$, then value of $f^{\prime }\left(x\right)$ is

1. $7$
2. $2f\left(x\right)+7$
3. $-3x$
4. $-3x+7$