For the given function, find the requested relative extrema or extreme value (without a calculator).y = xe4x; relative extrema

To find the relative extrema of the function y = xe^4x, we first need to find its derivative using the product rule. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first.

Step 1: Find the derivative of y = xe^4x The derivative of x is 1 and the derivative of e^4x is 4e^4x. Using the product rule, we get:

y' = 1 * e^4x + x * 4e^4x y' = e^4x + 4xe^4x

Step 2: Set the derivative equal to zero and solve for x 0 = e^4x + 4xe^4x 0 = e^4x(1 + 4x)

Setting each factor equal to zero gives x = 0 and x = -1/4.

Step 3: Use the second derivative test to determine whether these points are relative extrema The second derivative of y = xe^4x is y'' = 16xe^4x + 8e^4x.

For x = 0, y'' = 0, which is inconclusive.

For x = -1/4, y'' = -4e, which is less than 0, indicating a relative maximum.

Therefore, the function y = xe^4x has a relative maximum at x = -1/4.