Definite Integral

The given equation is:

We need to determine which of the following statements are TRUE about .

Step 1: Variable Substitution

Let's start by making a substitution to simplify the integral. Let: Therefore: Now, the differential becomes: With this substitution, the integral transforms into: Simplifying: This integral now depends on the value of .

Step 2: Analyzing the Integral

The integral's behavior depends on the choice of . We are asked to determine which values of make the integral equal to 1.

Step 3: Evaluating the Statements

Let's now analyze the given options.

Option A: "No satisfies the above equation"

This option is incorrect because there are values of that satisfy the equation.

Option B: "An integer satisfies the above equation"

This option is incorrect because no integer values of will satisfy the equation.

Option C: "An irrational number satisfies the above equation"

This is correct. Through analysis, we can find that irrational values of satisfy the equation.

Option D: "More than one satisfies the above equation"

This is also correct. There are indeed multiple values of that satisfy the equation.

Final Answer:

The correct options are: C, D.

Given: The function is defined as:

The first derivative is:

Step 1: Finding Critical Points

For the maximum or minimum, we set .

Step 2: Checking Behavior Around

We examine the sign of around . We can see that:

Conclusion

Since the derivative changes from negative to positive as crosses 0, has a minimum at .

Final Answer:

The function has a minimum at , and .

Mathematical Solution Using Logarithms

The function is defined as:

We can rearrange this equation to express in terms of :

The inverse of is given by:

Now, integrating:

This becomes:

Now simplifying:

Therefore:

Finally: