Final answer:
To find the critical numbers of the function f(x) = x^6e^(-8x), we need to find the values of x where the derivative of the function is equal to zero or undefined. The derivative of f(x) is 6x^5e^(-8x) + x^6(-8)e^(-8x). The only critical number of the function is x = 0.
Step-by-step explanation:
To find the critical numbers of the function f(x) = x^6e^(-8x), we need to find the values of x where the derivative of the function is equal to zero or undefined.
To find the derivative of f(x), we can use the product rule:
f'(x) = 6x^5e^(-8x) + x^6(-8)e^(-8x)
Setting this derivative equal to zero and solving for x, we find that x = 0 is a critical number. There are no other critical numbers since the derivative is never undefined for any value of x. Therefore, the only critical number of the function f(x) = x^6e^(-8x) is x = 0.