Suppose the function 
f(x) = 2 × 3x − 3.15x
describes a physical situation that makes sense only for whole numbers between 0 and 15. For what value of x does f reach a maximum, and what is that maximum value? (Suggestion: We suggest beginning with a table starting at 0 with a table increment of 1 and then panning further down the table. Round your answer for the maximum value to two decimal places.)

We are given the function:

\[
f(x) = 2 \times 3^x - 3.15^x
\]

where \(x\) can only take whole numbers between 0 and 15. Our task is to determine for which value of \(x\) the function reaches its maximum value, and to compute that maximum.

To solve this, we can:

1. Compute the value of the function for each \(x\) from 0 to 15.
2. Identify the \(x\) that gives the maximum value of \(f(x)\).

Let's calculate the values for \(f(x)\).The function \( f(x) = 2 \times 3^x - 3.15^x \) reaches its maximum value when \( x = 13 \). The maximum value of the function is approximately \( 182312.20 \).

Would you like more details or explanations on this result?

Here are some related questions to explore further:

1. What happens to the function as \( x \) approaches 15 or higher?
2. How does changing the base of the exponent affect the behavior of \( f(x) \)?
3. Can the function have other local maxima within a different range of \( x \)?
4. How would the function behave if the coefficients were altered (e.g., instead of 2 and 3.15)?
5. What physical scenario could this function represent?

**Tip**: When dealing with exponential functions, slight changes in the base can drastically affect the function's growth.

Suppose the function f(x) = 2 × 3^x − 3.15^x describes a physical situation that makes sense only for whole numbers between 0 and 15. For what value of x does f reach a maximum, and what is that maximum value? (Suggestion: We suggest beginning with a table starting at 0 with a table increment of 1 and then panning further down the table. Round your answer for the maximum value to two decimal places.)

Explore how to find the maximum value of the exponential function f(x) = 2 × 3^x − 3.15^x for whole numbers between 0 and 15. The solution includes a step-by-step approach to identifying the value of x where the function reaches its maximum, with the maximum value rounded to two decimal places. Suitable for high school students in Grades 10-12.

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