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. $19$
• B. $9$
• C. $17$
• D. $14$

Answer 1

The correct option is A $19$

The derivative will be,$f^{\prime }\left(x\right)-2f^{\prime }\left(2x\right)$
$f^{\prime }\left(x\right)-2f^{\prime }\left(2x\right){|}_1=5=f^{\prime }\left(1\right)-2f^{\prime }\left(2\right)...........\left(1\right)$
$f^{\prime }\left(x\right)-2f^{\prime }\left(2x\right){|}_2=7=f^{\prime }\left(2\right)-2f^{\prime }\left(4\right)$
Hence, $14=2f^{\prime }\left(2\right)-4f^{\prime }\left(4\right)$ ........................(2)
Adding (1) and (2),
$19=2f^{\prime }\left(2\right)-4f^{\prime }\left(4\right)+f^{\prime }\left(1\right)-2f^{\prime }\left(2\right)$
$\Rightarrow f^{\prime }\left(1\right)-4f^{\prime }\left(4\right)=19$