Question

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

• A. 2
• B. 3
• C. $\frac{1}{2}$
• D. 1

Answer 1

The correct option is A 2

$\text{Let}h\left(x\right)=f\left(x\right)-kg\left(x\right)\text{and}{h}^{\prime }\left(c\right)=0$
So Rolle's theorem is applicable.
$\Rightarrow h\left(0\right)=h\left(2\right)$
$\Rightarrow 3-k=5-2k\Rightarrow k=2$