How do you find the critical numbers of # f(t)=t(4-t)^(1/2)#?

Question

Evelyn Agard · Accepted Answer

Please see below.

Domain of #f# is #(-oo,4]# #f'(t) = (8-3t)/(2sqrt(4-t))# #f'(t) = 0# at #t=8/3# and #f'(t)# does not exist at #x=4#. Both numbers are in the domain of #f#, so both are critical numbers for #f#.

Scarlett Allison · Answer

undefined To find the critical numbers of the function $ f(t) = t(4 - t)^{\frac{1}{2}} $, we need to find the values of $ t $ where the derivative of the function equals zero or is undefined.

1. Find the derivative of $ f(t) $ using the product rule.
   $$
   f'(t) = (4 - t)^{\frac{1}{2}} + t \left( \frac{1}{2}(4 - t)^{-\frac{1}{2}} (-1) \right)
   $$
   Simplify this expression.

2. Set the derivative equal to zero and solve for $ t $.
   $$
   (4 - t)^{\frac{1}{2}} + t \left( \frac{1}{2}(4 - t)^{-\frac{1}{2}} (-1) \right) = 0
   $$

3. Solve for $ t $ in the above equation.

4. Check for any critical numbers by also considering where the derivative is undefined. In this case, the function is defined for all real numbers, so there are no points where the derivative is undefined.

5. The values of $ t $ obtained in step 3 are the critical numbers of the function $ f(t) $.