Final answer:
To find the critical numbers of the function f(x) = x^4(7x+4)^2/3, we need to find the values of x where the derivative of the function is equal to zero or undefined. The critical numbers are x = -4/7 and x = -26/21.
Step-by-step explanation:
To find the critical numbers of the function f(x) = x4(7x+4)2/3, we need to find the values of x where the derivative of the function is equal to zero or undefined. First, we find the derivative of f(x) using the power rule and chain rule: f'(x) = 4x3(7x+4)-1/3 * (3(7x+4) + (2/3)(7)) = 4x3(7x+4)-1/3 * (21x+12+14)/3 = 4x3(7x+4)-1/3 * (21x+26)/3
Next, we set the derivative equal to zero and solve for x: 4x3(7x+4)-1/3 * (21x+26)/3 = 0. Since the derivative is zero when the numerator is zero, we have two critical numbers: x = -4/7 and x = -26/21.