Question

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

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

Hint:

The slope of the normal is the negative reciprocal of the slope of the tangent. Use the angle of the normal to find the slope of the tangent.

Answer 1

The Correct Option is C

Step 1: Understanding the relationship between slope and angle.
The slope of the normal is the negative reciprocal of the slope of the tangent, and the slope of the normal is given by \( \tan\left( \frac{3\pi}{4} \right) \). The slope of the tangent is then \( f'(3) \).

Step 2: Conclusion.
Thus, the correct value of \( f'(3) \) is 1, so the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) 1}} \]