Application of Derivatives 1 Question 12
12. If the normal to the curve $y=f(x)$ at the point $(3,4)$ makes an angle $\frac{3 \pi}{4}$ with the positive $X$-axis, then $f^{\prime}(3)$ is equal to
(2000, 1M)
(a) -1
(b) $-3 / 4$
(c) $4 / 3$
(d) 1
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Answer:
Correct Answer: 12. $y+x-1=0$
Solution:
- Slope of tangent $y=f(x)$ is $\frac{d y}{d x}=f^{\prime}(x) _{(3,4)}$
Therefore, slope of normal
$$ \begin{array}{rlrl} & & =-\frac{1}{f^{\prime}(x) _{(3,4)}} & =-\frac{1}{f^{\prime}(3)} \\ \text { But } & -\frac{1}{f^{\prime}(3)} & =\tan \frac{3 \pi}{4} \\ \Rightarrow & \frac{-1}{f^{\prime}(3)} & =\tan \frac{\pi}{2}+\frac{\pi}{4}=-1 \\ \therefore & f^{\prime}(3) & =1 \end{array} $$
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