Question

Let $f(x)$ and $g(x)$ be two differentiable function in $R$ and $f(2)=8, g(2)=0, f(4)=10$ and $g(4)=8$ then

• A. $g ^{\prime}( x )>4 f ^{\prime}( x ) \quad \forall x \in(2,4)$
• B. $3 g ^{\prime}( x )=4 f ^{\prime}( x )$ for at least one $x \in(2,4)$
• C. $g ( x )> f ( x ) \quad \forall x \in(2,4)$
• D. $g^{\prime}(x)=4 f^{\prime}(x)$ for at least one $x \in(2,4)$

Answer 1

Answer: (D)

Consider h $(x)=g(x)-4 f(x)$, in $[2,4]$ also $h(2)=g(2)-4 f(2)=-32 ; h(4)=-32$
$\Rightarrow h ^{\prime}( x )=0$ for atleast one $x \in(2,4)$ using Rolle's theorem