Question

Let y=at^(2)e^(-bt) with a and b positive constants. for t>=0, what value of t maximizes y?

Answer 1

To find the value of t that maximizes y = at^2e^(-bt) with a and b positive constants, we need to take the derivative of y with respect to t, set it equal to zero, and solve for t.

Step-by-step explanation:

To find the value of t that maximizes y = at2e-bt, we need to find the maximum of the function. We can do this by taking the derivative of y with respect to t, setting it equal to zero, and solving for t.

First, let's find the derivative of y:

y' = 2ate-bt - abt2e-bt

Next, set the derivative equal to zero and solve for t:

2ate-bt - abt2e-bt = 0

2atebt = abt2

2ebt = bt

This equation cannot be solved algebraically, but it can be solved numerically using methods like Newton's method or the bisection method.