Find the following, rounded to two decimal places. (Enter your answers using interval notation.)

Given the function [tex] y = x \cdot 5^x \):

(a) The intervals on which the function is increasing or decreasing:
- Increasing: [tex] \square \)
- Decreasing: [tex] \square \)

(b) The range of the function:
[tex] \square \)



Answer :

Let's analyze the function \( y = x \cdot 5^x \) and address each part of the question step by step.

### Part (a) - Intervals of Increasing and Decreasing

To determine the intervals on which the function is increasing or decreasing, we need to find the first derivative of the function \( y \) and identify the critical points.

1. First Derivative:
The first derivative of the function \( y = x \cdot 5^x \) is found by using the product rule:
[tex]\[ y' = \frac{d}{dx} \left( x \cdot 5^x \right) = 5^x + x \cdot 5^x \ln(5) \][/tex]
Simplifying further:
[tex]\[ y' = 5^x (1 + x \ln(5)) \][/tex]

2. Critical Points:
To find the critical points, we set the first derivative equal to zero:
[tex]\[ 5^x (1 + x \ln(5)) = 0 \][/tex]
Since \( 5^x \) is never zero, we focus on:
[tex]\[ 1 + x \ln(5) = 0 \implies x = -\frac{1}{\ln(5)} \][/tex]

Thus, the critical point is:
[tex]\[ x = -\frac{1}{\ln(5)} \][/tex]

3. Intervals of Increase and Decreasing:
We need to test intervals around the critical point to determine whether the function is increasing or decreasing.

- For \( x < -\frac{1}{\ln(5)} \):
Choose a point slightly less than \( -\frac{1}{\ln(5)} \).
- If substituted into the derivative \( y' \), the resulting value is negative, showing that the function is decreasing in this interval.

- For \( x > -\frac{1}{\ln(5)} \):
Choose a point slightly greater than \( -\frac{1}{\ln(5)} \).
- If substituted into the derivative \( y' \), the resulting value is positive, showing that the function is increasing in this interval.

Therefore, the intervals are:
- Increasing: \( \left( -\frac{1}{\ln(5)}, \infty \right) \)
- Decreasing: \( \left( -\infty, -\frac{1}{\ln(5)} \right) \)

### Part (b) - Range of the Function

To determine the range of the function, we look at the behavior of \( y = x \cdot 5^x \) as \( x \) approaches positive and negative infinity.

1. As \( x \to +\infty \):
[tex]\[ y = x \cdot 5^x \to \infty \][/tex]

2. As \( x \to -\infty \):
[tex]\[ y = x \cdot 5^x \to 0 \][/tex]
This is because \( 5^x \) approaches \( 0 \) faster than \( x \) approaches \(-\infty\).

Combining these details, the range of the function is:
[tex]\[ \left( 0, \infty \right) \][/tex]

### Final Answers:
(a) Intervals on which the function is:
- Increasing: \( \left( -\frac{1}{\ln(5)}, \infty \right) \)
- Decreasing: \( \left( -\infty, -\frac{1}{\ln(5)} \right) \)

(b) The range of the function:
[tex]\[ (0, \infty) \][/tex]