Exponential And Logarithmic Functions

Given the function:

We need to find the derivative of this function. To do so, we apply logarithmic differentiation for the first term and the quotient rule for the second term:

Simplifying further:

Now, we need to determine where . To analyze the sign of , we find the common denominator for the fractions:

Expanding the numerator:

Simplifying the numerator:

Now, let's analyze the sign of based on the numerator and denominator. For the numerator, is always non-negative. For the denominator, and are also always non-negative, except when .

Therefore, is positive (greater than zero) when .

Hence, the function is increasing on the interval , which corresponds to option (B) .

Let Take natural logarithm: Differentiate both sides with respect to : LHS: RHS: So, Multiply both sides by : Now, simplify: Rewriting: Now evaluate at . First, compute : Try : Try again with Guess: Let’s take , try if it satisfies Let’s take Then: Let’s define: We discard 0, so: So, Now at : Hence,