Question

If the tangent to the function \( y = f(x) \) at \( (3, 4) \) makes an angle of \( \frac{3\pi}{4} \) with the positive direction of the x-axis in anticlockwise direction, then \( f'(3) \) is

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

Hint:

The slope of the tangent is the derivative of the function, and the angle the tangent makes with the x-axis can be used to find the derivative.

Answer 1

The Correct Option is A

Step 1: Using the Tangent Formula.
The slope of the tangent to the curve at a point is equal to the derivative of the function at that point. The angle \( \theta \) the tangent makes with the positive x-axis is related to the slope by: \[ \tan \theta = f'(x) \] For \( \theta = \frac{3\pi}{4} \), we get \( \tan \left( \frac{3\pi}{4} \right) = -1 \). Therefore, \( f'(3) = -1 \).
Step 2: Conclusion.
The correct answer is (A), \( -1 \).