Does the graph of the function y=3^3√x−1−−−−−show that the function is increasing or decreasing?

Question

Does the graph of the function  y=3^3√x−1−−−−−show that the function is increasing or decreasing?

Option 1: increasing

Option 2: decreasing

(1 point)
The correct answer is option 
.

GPT-4o mini · Answer

To determine whether the function $ y = 3\sqrt[3]{x} - 1 $ is increasing or decreasing, we should analyze its derivative. 1. **Find the derivative of the function**: To find the derivative $ y' $: $$ y = 3\sqrt[3]{x} - 1 = 3x^{1/3} - 1 $$ Applying the power rule, the derivative is: $$ y' = 3 \cdot \frac{1}{3} x^{-2/3} = x^{-2/3} $$ Thus, $$ y' = \frac{1}{x^{2/3}} $$ 2. **Analyze the sign of the derivative**: The expression $ \frac{1}{x^{2/3}} $ is positive for all $ x > 0 $ (since $ x^{2/3} $ is positive). 3. **Conclusion**: Since the derivative is positive for $ x > 0 $, the function is **increasing** on that interval. 4. **Final answer**: The correct answer is option **1: increasing**.