find the critical numbers of the function. (enter your answers as a comma - separated list. if an answer…

Explanation:

Step1: Differentiate the function

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (h(t)=t^{3/4}-3t^{1/4}), we have (h^\prime(t)=\frac{3}{4}t^{\frac{3}{4}-1}-3\times\frac{1}{4}t^{\frac{1}{4}-1}=\frac{3}{4}t^{-\frac{1}{4}}-\frac{3}{4}t^{-\frac{3}{4}}). Simplify (h^\prime(t)=\frac{3}{4t^{1/4}}-\frac{3}{4t^{3/4}}=\frac{3t - 3}{4t^{3/4}}) (by getting a common - denominator (4t^{3/4})).

Step2: Find where (h^\prime(t) = 0) or (h^\prime(t)) is undefined

• Set (h^\prime(t)=0): Set (\frac{3t - 3}{4t^{3/4}} = 0). Since the denominator (4t^{3/4}\neq0) for (t>0), we solve (3t - 3 = 0). (3t-3 = 0) gives (t = 1) (add 3 to both sides: (3t=3), then divide by 3).
• Find where (h^\prime(t)) is undefined: The function (h^\prime(t)) is undefined when (t = 0) (because the denominator (4t^{3/4}=0) when (t = 0)). But we also need to check the domain of the original function (h(t)). The domain of (h(t)=t^{3/4}-3t^{1/4}=\sqrt[4]{t^3}-3\sqrt[4]{t}) is (t\geq0).

Answer:

(0,1)