Final answer:
The function y = bxe^(-ax) with a local maximum at (3,2) is found by setting the first derivative equal to zero at x = 3 and ensuring the function yields a value of 2 at x = 3. The values a = 1 and b = 2e^3 satisfy this, producing the function y = 2xe^(-x), which corresponds to option C.
Step-by-step explanation:
We are asked to find a formula for a function of the form y = bxe^(-ax) with a local maximum at (3,2). The presence of a local maximum implies that the derivative of the function will be zero at that point.
Let's take the first derivative of the general form y = bxe^(-ax) with respect to x:
y' = b * e^(-ax) + b * x * (-a) * e^(-ax) = be^(-ax) - abxe^(-ax).
To find the values of a and b, we need to satisfy two conditions:
- The derivative y' is zero at x = 3.
- The function itself is equal to 2 when x = 3.
By plugging in x = 3 into the first derivative and setting it to zero, and plugging x = 3 into the function and setting it equal to 2, we can solve for a and b. After calculations, we find that a = 1 and b = 2e^3 which simplifies the function to y = 2xe^(-x). Therefore, the correct choice that fits these conditions is C. y = 2xe^(-3x).