Question

Hard question!
Find the critical # of the function.
g(t)=|6t-9|
t=

Answer 1

\[g(t)=|6t-9|=\begin{cases}6t-9&\text{for }t>\frac{3}{2}\\\\0&\text{for }t=\frac{3}{2}\\\\9-6t&\text{for }t<\frac{3}{2}\end{cases}\]
(This follows from the definition of the absolute value function.)

Differentiating \(g\), we get
\[g'(t)=\begin{cases}6&\text{for }t>\frac{3}{2}\\\\DNE&\text{for }t=\frac{3}{2}\\\\-6&\text{for }t<\frac{3}{2}\end{cases}\]
Critical points occur at the \(t\) values that make \(g'(t)\) zero or undefined.