Find the largest interval on which the given function is increasing.


c. g(x) = (3x - 1)¹/³

Verified step by step guidance

1. To determine where the function \( g(x) = (3x - 1)^{1/3} \) is increasing, we first need to find its derivative. The derivative will help us understand the behavior of the function.
2. Apply the chain rule to differentiate \( g(x) = (3x - 1)^{1/3} \). The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).
3. Let \( u = 3x - 1 \). Then \( g(x) = u^{1/3} \). The derivative of \( u^{1/3} \) with respect to \( u \) is \( \frac{1}{3}u^{-2/3} \).
4. Now, find \( \frac{du}{dx} \), which is the derivative of \( u = 3x - 1 \) with respect to \( x \). This is simply \( 3 \).
5. Combine these results using the chain rule: \( g'(x) = \frac{1}{3}(3x - 1)^{-2/3} \cdot 3 \). Simplify this expression to find where \( g'(x) > 0 \), which will give the interval where the function is increasing.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus for determining the behavior of functions, including identifying intervals where a function is increasing or decreasing. If the derivative is positive over an interval, the function is increasing on that interval.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for analyzing the function's behavior, as they can indicate potential local maxima, minima, or points of inflection. To find intervals of increase or decrease, one must evaluate the derivative at these critical points and test the sign of the derivative in the intervals they create.

Increasing and Decreasing Intervals

An interval is considered increasing if the function's output rises as the input increases, which corresponds to the derivative being positive. Conversely, a function is decreasing when its output falls as the input rises, indicated by a negative derivative. By analyzing the sign of the derivative across different intervals, one can determine where the function is increasing or decreasing.