Question

If $f\left(x\right)=(x-a)g\left(x\right)$ and g(x) is continuous at $x=a$ then $f^{\prime }\left(1\right)$=?

• A. $1$
• B. $g^{\prime }\left(1\right)$
• C. $g\left(1\right)$
• D. $a{g}^{\prime }\left(1\right)$

Answer 1

The correct option is C $g\left(1\right)$

Given function:

$f\left(x\right)=(x-a)g\left(x\right)$

On differentiating with respect to $x$, we get

$f^{\prime }\left(x\right)=(x-a)g^{\prime }\left(x\right)+g\left(x\right)$

Since $g\left(x\right)$ is continuous at $x=a$,

So $g^{\prime }\left(x\right)=0$ at $x=a$

Hence,

$f^{\prime }\left(x\right)=g\left(x\right)$

Substituting $x=1$, we get

$f^{\prime }\left(1\right)=g\left(1\right).$

This is the required solution.