Does the graph of the function y=3x−1−−−−√3

Question

show that the function is increasing or decreasing?

Option 1: increasing

Option 2: decreasing

(1 point)
The correct answer is option
.

Answer 1

To determine whether the function \( y = \sqrt{3x - 1} \) is increasing or decreasing, we can find its derivative.

1. Differentiate the function: \[ y = \sqrt{3x - 1} \] Using the chain rule: \[ \frac{dy}{dx} = \frac{1}{2}(3x - 1)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x - 1}} \]

2. Analyze the derivative: The expression \( \sqrt{3x - 1} \) is positive for \( x > \frac{1}{3} \), since it is only defined for \( 3x - 1 \geq 0 \). Therefore, the derivative \( \frac{dy}{dx} \) will also be positive for all \( x > \frac{1}{3} \).

Since the derivative is positive for \( x > \frac{1}{3} \), the function is increasing in that interval.

Thus, the correct answer is:

Option 1: increasing.