Theory and Examples


In Exercises 51–54,


d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)


y = x³/3

Verified step by step guidance

1. To determine where the function y = f(x) = x³/3 is increasing or decreasing, we first need to find its derivative. The derivative, f'(x), will help us understand the behavior of the function.
2. Calculate the derivative of y = x³/3 with respect to x. Using the power rule, the derivative of x³ is 3x². Therefore, f'(x) = (3x²)/3 = x².
3. Analyze the sign of the derivative f'(x) = x². Since x² is always non-negative (x² ≥ 0 for all x), the derivative is zero at x = 0 and positive for all other x-values.
4. Determine the intervals of increase and decrease: Since f'(x) = x² is positive for all x ≠ 0, the function y = x³/3 is increasing on the intervals (-∞, 0) and (0, ∞). There are no intervals where the function is decreasing because the derivative is never negative.
5. Relate this to part (c): The critical point found in part (c) is x = 0, where the derivative is zero. This is a point of inflection, not a local maximum or minimum, as the function changes concavity but continues to increase on either side of x = 0.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input changes. For a function y = f(x), the derivative f'(x) provides the rate of change of y with respect to x. In this context, finding the derivative of y = x³/3 helps determine where the function is increasing or decreasing by analyzing the sign of f'(x).

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential locations where the function changes from increasing to decreasing or vice versa. For y = x³/3, finding the critical points involves solving f'(x) = 0, which helps identify intervals of increase or decrease.

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if the derivative is negative. By analyzing the sign of the derivative f'(x) for y = x³/3, we can determine the intervals over which the function increases or decreases, providing insight into the function's behavior.