5. Consider the differential equation from question 4: $$ \begin{array}{c} -1(-1-4) \ -1(-5) \ 5 \end{array} $$ For which values of $$ y $$ is a solution $$ y(t) $$ increasing? $$ \begin{array}{c} x(2-4) \ 2(-2) \ -4 \end{array} $$ (a) $$ y4 $$ only (c) $$ y4 $$ (d) $$ 0

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5. Consider the differential equation from question 4: $$ \begin{array}{c} -1(-1-4) \ -1(-5) \ 5 \end{array} $$ For which values of $$ y $$ is a solution $$ y(t) $$ increasing? $$ \begin{array}{c} x(2-4) \ 2(-2) \ -4 \end{array} $$ (a) $$ y<0 $$ only (b) $$ y>4 $$ only (c) $$ y<0, y>4 $$ (d) $$ 0

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To determine for which values of $$ y $$ the solution $$ y(t) $$ is increasing, let's analyze the given differential equation and the provided evaluations. ### Given Information: The differential equation from question 4 appears to be: $$ y' = y(y - 4) $$ This can be inferred from the evaluations provided: $$ \begin{array}{c} y = -1: \quad y' = -1(-1 - 4) = 5 \ y = 2: \quad y' = 2(2 - 4) = -4 \ \end{array} $$ ### Analyzing $$ y' = y(y - 4) $$: The sign of $$ y' $$ determines whether $$ y(t) $$ is increasing or decreasing: - **Increasing** when $$ y' > 0 $$ - **Decreasing** when $$ y' < 0 $$ Let's find the intervals where $$ y' > 0 $$: $$ y(y - 4) > 0 $$ This inequality holds true in two scenarios: 1. **Both factors are positive:** $$ y > 0 \quad \text{and} \quad y - 4 > 0 \quad \Rightarrow \quad y > 4 $$ 2. **Both factors are negative:** $$ y < 0 \quad \text{and} \quad y - 4 < 0 \quad \Rightarrow \quad y < 0 $$ ### Conclusion: The solution $$ y(t) $$ is increasing for values of $$ y $$ where $$ y < 0 $$ **or** $$ y > 4 $$. **Answer:** (c) $$ y < 0, \ y > 4 $$ The solution $$ y(t) $$ is increasing when $$ y < 0 $$ or $$ y > 4 $$.

Consider the differential equation from question 4:

For which values of is a solution increasing?

(a) only
(b) only
©
(d)

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Consider the differential equation from question 4: For which values of is a solution increasing? (a) only (b) only © (d)