Question

Find the critical numbers of the function h(t) = t^(3/4) - 3t^(1/4).

Answer 1

The critical number of the function h(t) = t^(3/4) - 3t^(1/4) is found by setting its first derivative, h'(t) = (3/4)t^(-1/4) - (3/4), equal to zero, which yields the critical number t = 1.

Step-by-step explanation:

Finding Critical Numbers of a Function

To find the critical numbers of the function h(t) = t3/4 - 3t1/4, we need to determine where its first derivative h'(t) is equal to zero or undefined. To differentiate the function, apply the power rule of derivatives:

h'(t) = (3/4)t-1/4 - (3/4)

We'll set this derivative equal to zero to find the critical points:

0 = (3/4)t-1/4 - (3/4)

By solving this equation for t, we find that the critical number is t = 1.

Note that the derivative is never undefined because the function and its derivative are defined for all t > 0. Thus, our only critical number is t = 1.