Question:

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. Step 1: Determine the slope of the normal. The slope of the normal, $m_n$, is given by the tangent of the angle it makes with the positive X-axis: $$m_n = \tan\left(\frac{3\pi}{4}\right) = -1$$ Step 2: Determine the slope of the tangent. The slope of the tangent, $f'(3)$, is related to the slope of the normal by the condition that their product is $-1$: $$f'(3) \cdot m_n = -1$$ $$f'(3) \cdot (-1) = -1$$ $$f'(3) = 1$$