Question

If the normal to the curve y = f(x) at (3, 4) makes an angle $\frac{3\mathrm{\pi }}{4}$ with positive x-axis, then f '(3) is equal to ________________.

Answer 1

It is given that, the normal to the curve y = f(x) at (3,4) makes an angle $\frac{3\mathrm{\pi }}{4}$ with positive x-axis.

∴ Slope of the normal = $\tan \frac{3\mathrm{\pi }}{4}=\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{4}\right)=-\tan \frac{\mathrm{\pi }}{4}$ = −1 .....(1)

Now, y = f(x)

$\Rightarrow \frac{dy}{dx}=f\text{'}\left(x\right)$

∴ Slope of the normal at (3, 4) $=-\frac{1}{{\left({\displaystyle \frac{dy}{dx}}\right)}_{\left(3,4\right)}}=-\frac{1}{f\text{'}\left(3\right)}$ .....(2)

From (1) and (2), we have

$-\frac{1}{f\text{'}\left(3\right)}=-1$

$\Rightarrow f\text{'}\left(3\right)=1$

Thus, the value of $f\text{'}\left(3\right)$ is 1.


If the normal to the curve y = f(x) at (3, 4) makes an angle $\frac{3\mathrm{\pi }}{4}$ with positive x-axis, then f '(3) is equal to ___1___.