17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.


b. In what regions are solutions increasing? Decreasing?


y'(t) = y(y+3)(4-y)

Verified step by step guidance

1. Identify the differential equation given: \(y'(t) = y(y+3)(4-y)\).
2. Determine the critical points by setting the right-hand side equal to zero: solve \(y(y+3)(4-y) = 0\). The solutions are \(y = 0\), \(y = -3\), and \(y = 4\).
3. Divide the real line into intervals based on these critical points: \((-\infty, -3)\), \((-3, 0)\), \((0, 4)\), and \((4, \infty)\).
4. Analyze the sign of \(y'(t)\) in each interval by choosing a test value from each interval and substituting it into \(y(y+3)(4-y)\) to determine if the product is positive or negative.
5. Conclude that solutions are increasing where \(y'(t) > 0\) and decreasing where \(y'(t) < 0\) based on the sign analysis in each interval.

Key Concepts

Sign of the Derivative and Monotonicity

The sign of the derivative y'(t) determines whether the solution y(t) is increasing or decreasing. If y'(t) > 0, the function is increasing; if y'(t) < 0, it is decreasing. Analyzing the sign of y'(t) over intervals helps identify where solutions rise or fall.

Factoring and Critical Points

Factoring the derivative expression y'(t) = y(y+3)(4-y) reveals critical points where y'(t) = 0, specifically at y = 0, y = -3, and y = 4. These points divide the y-axis into intervals where the sign of y'(t) can be tested to determine increasing or decreasing behavior.

Interval Testing for Sign Analysis

To determine where y'(t) is positive or negative, select test values from each interval defined by the critical points. Substituting these values into y'(t) shows the sign of the derivative, which indicates whether the solution is increasing or decreasing in that region.