Final answer:
The critical numbers of the function f(x) = x^(4/5)(x-4)^2 are found by differentiating the function, setting the derivative equal to zero, and solving the resulting equation. This involves using the product rule, chain rule, and algebraic methods.
Step-by-step explanation:
To find the critical numbers of the function f(x) = x4/5(x-4)2, we need to look for values of x where the first derivative of f(x) is either zero or undefined. The first step is to differentiate the function with respect to x.
Let's find the derivative of f(x):
f'(x) = d/dx [x4/5(x-4)2]
This requires the product rule and the chain rule.
Using the product rule, we get:
This simplifies to:
f'(x) = (4/5)x-1/5(x-4)2 + 2x4/5(x-4).
To find the critical numbers, we set the derivative equal to zero:
(4/5)x-1/5(x-4)2 + 2x4/5(x-4) = 0.
We can solve this equation by factoring or using algebraic methods. The critical numbers are the solutions to this equation, excluding any that might make the original function undefined.