Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = x³/8

Verified step by step guidance

1. First, identify the function given: \( y = \frac{x^3}{8} \). This is a cubic function, which is generally of the form \( y = ax^3 + bx^2 + cx + d \). In this case, \( a = \frac{1}{8} \), \( b = 0 \), \( c = 0 \), and \( d = 0 \).
2. To determine the intervals of increase and decrease, find the first derivative of the function. The derivative \( y' \) of \( y = \frac{x^3}{8} \) is \( y' = \frac{3x^2}{8} \).
3. Analyze the first derivative \( y' = \frac{3x^2}{8} \). Since \( 3x^2 \) is always non-negative and \( \frac{1}{8} \) is positive, \( y' \geq 0 \) for all \( x \). This indicates that the function is non-decreasing everywhere.
4. Since the derivative is zero only at \( x = 0 \), the function is constant at this point and increasing elsewhere. Therefore, the function is increasing on the intervals \( (-\infty, 0) \) and \( (0, \infty) \).
5. Check for symmetries: The function \( y = \frac{x^3}{8} \) is an odd function because \( f(-x) = -f(x) \). This means the graph is symmetric with respect to the origin.

Key Concepts

Increasing and Decreasing Functions

A function is increasing on an interval if, for any two numbers in that interval, a larger input results in a larger output. Conversely, it is decreasing if a larger input results in a smaller output. To determine these intervals, one typically examines the sign of the derivative: positive for increasing and negative for decreasing.

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph is invariant under certain transformations, such as reflection or rotation. Common types include even symmetry (y-axis symmetry) and odd symmetry (origin symmetry). For the function y = x³/8, checking for odd symmetry involves verifying if f(-x) = -f(x).

Derivative and Critical Points

The derivative of a function provides information about its rate of change. Critical points occur where the derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. For y = x³/8, the derivative y' = (3/8)x² helps identify intervals of increase or decrease by analyzing where it is positive or negative.