Final answer:
To find the critical points of the function f(x) = x^(2/3)(x - 5), calculate its derivative and set it equal to zero or look for points where it's undefined. Solve for x in the resulting equations to get the critical points.
Step-by-step explanation:
To find the critical points of the function f(x) = x^(2/3)(x - 5), we need to find the values of x where the first derivative of the function is zero or undefined. Taking the derivative, we apply the product rule:
f'(x) = d/dx [x^(2/3)](x - 5) + x^(2/3)d/dx [x - 5]
= (2/3)x^(-1/3)(x - 5) + x^(2/3)(1)
Setting the derivative equal to zero gives us:
(2/3)x^(-1/3)(x - 5) + x^(2/3) = 0
This equation is true when x^(2/3) = 0 or (2/3)x^(-1/3)(x - 5) + 1 = 0. Solving x^(2/3) = 0 gives x = 0. To solve the second equation, we can multiply through by x^(1/3) to clear the fraction and then solve the resulting quadratic equation. The critical points occur where f'(x) is zero or undefined, which are the solutions to these equations.