xe^(4x)
List all the critical numbers:
To find the critical numbers of a function, we need to find the values of x for which the derivative of the function is either zero or undefined.
Let's take the given function, f(x) = xe^(4x), and find its derivative.
The derivative of f(x) can be found using the product rule, which states that if we have a function h(x) = u(x)v(x), then the derivative of h(x) is given by h'(x) = u'(x)v(x) + u(x)v'(x).
Using the product rule, we can find the derivative of f(x) = xe^(4x) as follows:
f'(x) = (1)(e^(4x)) + (x)((4)e^(4x))
= e^(4x) + 4xe^(4x)
Now, we need to find the critical numbers by setting the derivative equal to zero and solving for x:
e^(4x) + 4xe^(4x) = 0
To solve this equation, we can factor out e^(4x):
e^(4x)(1 + 4x) = 0
From here, we have two possibilities:
1) e^(4x) = 0: This equation has no solutions since e^(4x) is always positive.
2) 1 + 4x = 0: Solving this equation for x, we get:
4x = -1
x = -1/4
Therefore, the only critical number of the function f(x) = xe^(4x) is x = -1/4.