Let's analyze the task step by step and solve it.

Task Breakdown:

1. The original problem asks to find intervals where the function $y = (x^2 - 3)e^{5 - x}$ is increasing.

   • We must determine the monotonic behavior of the function $y(x)$.

2. Steps:

   • Find the derivative $y'(x)$ to determine where $y(x)$ is increasing or decreasing.
   • Solve $y'(x) > 0$ for increasing intervals.
   • Use the provided intervals to identify valid monotonicity regions.

Step 1: Find the derivative $y'(x)$

The function is $y = (x^2 - 3)e^{5 - x}$. Applying the product rule:

[ y'(x) = \frac{d}{dx}[(x^2 - 3)] \cdot e^{5 - x} + (x^2 - 3) \cdot \frac{d}{dx}[e^{5 - x}] ]

• The derivative of $x^2 - 3$ is $2x$.
• The derivative of $e^{5 - x}$ is $-e^{5 - x}$ (chain rule).

Substitute these into the product rule:

$y'(x) = 2x e^{5 - x} + (x^2 - 3)(-e^{5 - x})$

Factor out $e^{5 - x}$:

$y'(x) = e^{5 - x} \big[ 2x - (x^2 - 3) \big]$

Simplify the term in brackets:

$y'(x) = e^{5 - x} \big[ -x^2 + 2x + 3 \big]$

Thus:

$y'(x) = e^{5 - x} (-x^2 + 2x + 3)$

Step 2: Analyze $y'(x)$

• The term $e^{5 - x} > 0$ for all $x$ because the exponential function is always positive. Therefore, the sign of $y'(x)$ depends on the quadratic expression $-x^2 + 2x + 3$.
• Factorize $-x^2 + 2x + 3$: $-x^2 + 2x + 3 = -(x^2 - 2x - 3)$ Factor the quadratic: $x^2 - 2x - 3 = (x - 3)(x + 1)$ Thus: $-x^2 + 2x + 3 = -(x - 3)(x + 1)$

So: $y'(x) = e^{5 - x} \cdot -(x - 3)(x + 1)$

Step 3: Solve $y'(x) > 0$

The derivative $y'(x) > 0$ when $-(x - 3)(x + 1) > 0$.

• Analyze the inequality $-(x - 3)(x + 1) > 0$: $(x - 3)(x + 1) < 0$
• Solve $(x - 3)(x + 1) < 0$ using a sign chart:
  • The critical points are $x = -1$ and $x = 3$.
  • Test intervals:
    • For $x < -1$: $(x - 3)(x + 1) > 0$ (negative result).
    • For $-1 < x < 3$: $(x - 3)(x + 1) < 0$ (positive result).
    • For $x > 3$: $(x - 3)(x + 1) > 0$ (negative result).

Thus, $y'(x) > 0$ on $(-1, 3)$.

Step 4: Compare with the given intervals

The problem asks to select intervals where $y(x)$ is increasing:

• $(-3, -2)$: Not in $(-1, 3)$.
• $(0, 1)$: Lies within $(-1, 3)$, so it is correct.
• $(2, 3)$: Lies within $(-1, 3)$, so it is correct.

Final Answer:

The intervals where $y(x)$ is increasing are:

$(0, 1) \quad \text{and} \quad (2, 3)$

Would you like me to explain any steps in more detail?

Here are 5 related questions for further understanding:

1. How do we interpret the signs of the derivative to determine increasing/decreasing behavior?
2. What is the significance of factoring the quadratic in the derivative?
3. How do we use a sign chart to solve inequalities?
4. Why is $e^{5 - x}$ always positive, and how does it affect the derivative?
5. How can we determine critical points for non-polynomial functions?

Tip: Always check the domain of your function and derivative before analyzing monotonic behavior.