Question

Suppose the function $f\left(x\right)-f\left(2x\right)$ has the derivative $5$ at $x=1$ and derivative $7$ at $x=2.$ The derivative of the function $f\left(x\right)-f\left(4x\right)$ at $x=1$ has the value equal to

• A. $9$
• B. $17$
• C. $14$
• D. $19$

Answer 1

The correct option is D $19$

$y=f\left(x\right)-f\left(2x\right)\Rightarrow y^{\prime }=f^{\prime }\left(x\right)-2f^{\prime }\left(2x\right)$
$\Rightarrow y^{\prime }\left(1\right)=f^{\prime }\left(1\right)-2f^{\prime }\left(2\right)=5\dots \left(1\right)$
and $y^{\prime }\left(2\right)=f^{\prime }\left(2\right)-2f^{\prime }\left(4\right)=7\dots \left(2\right)$
Multiplying equation $\left(2\right)$ by $2$ and adding with equation $\left(1\right),$
$f^{\prime }\left(1\right)-4f^{\prime }\left(4\right)=19\dots \left(3\right)$

Now, let $g\left(x\right)=f\left(x\right)-f\left(4x\right)$
$\Rightarrow g^{\prime }\left(x\right)=f^{\prime }\left(x\right)-4f^{\prime }\left(4x\right)$
$\Rightarrow g^{\prime }\left(1\right)=f^{\prime }\left(1\right)-4f^{\prime }\left(4\right)$
From equation $\left(3\right),$
$g^{\prime }\left(1\right)=19$