Increasing And Decreasing Functions

Given function:

Derivative:

To find when the function is strictly increasing, we solve:

Now analyze:

The inequality holds in the intervals:

At , , so the derivative is zero — not strictly increasing at that point. And for , , meaning , so the function is decreasing there.

✅ Final Answer:
The function is strictly increasing on the interval:

To determine the behavior of the function , we consider its derivative to analyze intervals of increase and decrease.

The derivative is:

.

1. Set the derivative equal to zero to find critical points:

⟹ .

2. Analyze the sign of on the intervals divided by this critical point:

- For , . Thus, is increasing.

- For , . Thus, is decreasing.

3. Conclusion:

is increasing in and decreasing in .

For the function to be decreasing for , its derivative must be non-positive for all . Compute the derivative:

For to be decreasing, we need:

for all

This implies:

Since the range of is , to satisfy for all , must be greater than or equal to the maximum value of :

Hence, for the function to be strictly decreasing, must be: