Question

If the normal to the curve $y=f\left(x\right)$ at the point $(3,4)$ makes an angle $\frac{3\pi }{4}$ with the positive x-axis then $f^{\prime }\left(3\right)$ is equal to

• A. $-1$
• B. $-\frac{3}{4}$
• C. $\frac{4}{3}$
• D. $1$

Answer 1

Answer: (D)

$1$

Slope of normal
$=\frac{-1}{f^{\prime }\left(x\right)}$
$=tan\left(\frac{3\pi }{4}\right)$
$=-1$
Hence
$f^{\prime }\left(x\right)=1$
Or
$\frac{dy}{dx}=1$
Or
$y=x+c$
Hence
$y=f\left(x\right)$ is an equation of a straight line parallel to $y=x$.
Hence
$f^{\prime }\left(x\right)$ is independent of x and its value is 1.