Applications Of Derivatives

Step 1: Understanding the Concept:
We first determine the function using functional equations, find its maximum value , and then solve the area-related integral to find .
Step 2: Detailed Explanation:
Given:
Replace with :

From (2), . Substitute into (1):

.
Maximum value .
Now, .
Curves and intersect at and .
Area
Area .
Equating Area to :
.
We need .
Step 3: Final Answer:
The value of is 16.

Step 1: Understanding the Concept:
To determine whether a function is increasing or decreasing, we evaluate the sign of its first derivative .
A function is increasing if and decreasing if .

Step 2: Key Formula or Approach:
Given , where .
Using the chain rule, .

Step 3: Detailed Explanation:
First, find :


Now, consider .
Given , is a strictly increasing function.
Since , and for all , it follows that:
.
The derivative of is:

Analyzing the interval :
In this range, , so .
Since and , we have .
Thus, is decreasing in . (Statement I is False)
Analyzing the interval :
In this range, , so .
Thus, .
So, is increasing in . (Statement II is False)

Step 4: Final Answer:
Both statements (I) and (II) are false.