Final answer:
The critical number for the function g(t) = 30t^(2/3) + 6t^(5/3) is t = -2.
Step-by-step explanation:
The critical numbers of a function are the values of the independent variable at which the derivative of the function is either zero or undefined. To find the critical numbers of the function g(t) = 30t^(2/3) + 6t^(5/3), we need to find the values of t for which the derivative of g(t) is zero or undefined.
Taking the derivative of g(t) using the power rule and simplifying, we get g'(t) = 20t^(-1/3) + 10t^(2/3). Setting g'(t) equal to zero and solving for t, we have 20t^(-1/3) + 10t^(2/3) = 0. Multiplying both sides by t^(1/3), we get 20 + 10t = 0. Subtracting 20 from both sides and dividing by 10, we find that t = -2. Plugging this value back into g(t), we get g(-2) = 30(-2)^(2/3) + 6(-2)^(5/3) = -48.
Therefore, the critical number for g(t) is t = -2.