Find any critical numbers of the function. (Enter your answers as a comma-separated list.)
g(t) = 3t ln t
To find the critical numbers of a function, we need to determine where the derivative of the function is equal to zero or undefined. Let's find the derivative of the function g(t) = 3t ln t first.
1. Start by using the product rule to differentiate the function g(t) = 3t ln t:
g'(t) = 3(ln t) + t(1/t)
g'(t) = 3(ln t) + 1
2. Now, set the derivative equal to zero and solve for t:
3(ln t) + 1 = 0
3(ln t) = -1
ln t = -1/3
3. Exponentiate both sides of the equation to get rid of the natural logarithm:
e^(ln t) = e^(-1/3)
t = e^(-1/3)
Therefore, the critical number of the function g(t) = 3t ln t is t = e^(-1/3).
Note: Since the derivative g'(t) = 3(ln t) + 1 is defined for all positive values of t, there are no values of t that make the derivative undefined.
first, just to see the big picture,
do you know what critical numbers are?