Increasing And Decreasing Functions

We are given the function .
Step 1: Find the derivative of }
The derivative of the function is:
Step 2: Find where }
For the function to be strictly decreasing, we need: This gives: Solving for : Thus, the function is strictly decreasing in the interval .

Therefore, the correct answer is (A) .

1. Understand the problem:

We need to determine the behavior of the function by analyzing where it is increasing or decreasing.

2. Find the first derivative:

Using the product rule and chain rule:

Simplify the expression:

3. Analyze the sign of :

Since for all real , the sign of depends on .

4. Solve :

Find the roots of the quadratic equation:

Roots: and .

The quadratic is a downward-opening parabola. It is positive between its roots:

5. Determine increasing/decreasing intervals:

The function is:

• Increasing on (since )
• Decreasing elsewhere (since for or )

Correct Answer: (D) Increasing in

1. Understand the problem:

We need to determine where the function (for ) is strictly increasing.

2. Use logarithmic differentiation:

Let . Take the natural logarithm:

Differentiate implicitly with respect to :

Thus:

3. Determine where :

Since for , the sign of depends on :

Correct Answer: (C)