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

Answer 1

Answer: (D)

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