Question

The normal to the curve \(y=f(x)\) at the point \((3,4)\) makes an angle \(\frac{3\pi}{4}\) with positive \(x\)-axis, then \(f'(3)=\)

• A. \(3\)
• B. \(2\)
• C. \(1\)
• D. \(4\)

Hint:

The slope of the tangent to \(y=f(x)\) at \(x=a\) is \(f'(a)\). If the normal has slope \(m_n\), then the tangent slope is \[ m_t=-\frac{1}{m_n}. \]

Answer 1

The Correct Option is C

Step 1: Find the slope of the normal.
The normal at \((3,4)\) makes angle \(3\pi/4\) with the positive x-axis. Slope of normal \(= \tan(3\pi/4) = -1\).

Step 2: Find the slope of the tangent.
Since tangent and normal are perpendicular: slope of tangent \(= -\dfrac{1}{-1} = 1\).

Step 3: Conclude.
\(f'(3) =\) slope of tangent at \(x=3\) \(= 1\).
\[ \boxed{f'(3) = 1} \]