Final answer:
To find the critical points of the function 5x^(2/3) - x^(5/3), we must differentiate the function and set the derivative equal to zero. The critical points are found to be x = 0 and x = 3.
Step-by-step explanation:
To find the critical points of the function 5x(2/3) - x(5/3), we need to find where the derivative of the function is equal to zero or does not exist. To do this, we start by differentiating the function with respect to x.
Let f(x) = 5x(2/3) - x(5/3). The derivative f'(x) is obtained using the power rule which states that d/dx of xn is n*x(n-1).
So,
f'(x) = d/dx [5x(2/3) - x(5/3)]
= 5 * (2/3) * x(-1/3) - (5/3) * x(2/3)
Now set the derivative equal to zero and solve for x:
5*(2/3)*x(-1/3) - (5/3)*x(2/3) = 0
3x(-1/3) - x(2/3) = 0
x(-1/3) * (3 - x) = 0
Thus, x = 0 or x = 3 are the critical points.