`(y)=2-3x-5x^(2)`

Options

A. `I_(1)=(2,3), I_(2)=(3,5)`
B. `I_(1)=(2,5), I_(2)=(3,4)`
C. `I_(1)=(infty, -3),I_(2)=(-3, infty)`
D. `I_(1)=(-infty, -0.3),I_(2)=(-0.3, infty)`

Answer

To determine the intervals where the function \( y = 2 - 3x - 5x^2 \) is increasing or decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first need to find its derivative \( \frac{dy}{dx} \).

Given:
\[ y = 2 - 3x - 5x^2 \]

Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = -3 - 10x \]

### Step 2: Set the derivative greater than zero for increasing intervals
To find where the function is increasing, we set the derivative greater than zero:
\[ -3 - 10x > 0 \]

### Step 3: Solve the inequality
Rearranging the inequality:
\[ -10x > 3 \]
Dividing both sides by -10 (remember to reverse the inequality sign):
\[ x < -\frac{3}{10} \]

### Step 4: Determine the interval for increasing function
The function is increasing for:
\[ x \in (-\infty, -0.3) \]

### Step 5: Set the derivative less than zero for decreasing intervals
Next, we set the derivative less than zero to find where the function is decreasing:
\[ -3 - 10x < 0 \]

### Step 6: Solve the inequality
Rearranging the inequality:
\[ -10x < 3 \]
Dividing both sides by -10 (again reversing the inequality sign):
\[ x > -\frac{3}{10} \]

### Step 7: Determine the interval for decreasing function
The function is decreasing for:
\[ x \in (-0.3, \infty) \]

### Summary of intervals:
- The function is increasing on the interval \( (-\infty, -0.3) \).
- The function is decreasing on the interval \( (-0.3, \infty) \).