I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 3.2 : Interpretation of the Derivative
11. Determine where, if anywhere, the function \(g\left( x \right) = {x^3} - 2{x^2} + x - 1\) stops changing.
Show SolutionWe know that the derivative of a function gives us the rate of change of the function and so we’ll first need the derivative of this function. We computed this derivative in Problem 7 from the previous section and so we won’t show the work here. If you need the practice you should go back and redo that problem before proceeding.
From our previous work (with a corresponding change of variables) we know that the derivative is,
\[g'\left( x \right) = 3{x^2} - 4x + 1\]If the function stops changing at a point then the derivative will be zero at that point. So, to determine if we function stops changing we will need to solve,
\[\begin{align*}g'\left( x \right) & = 0\\ 3{x^2} - 4x + 1 & = 0\\ \left( {3x - 1} \right)\left( {x - 1} \right) & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}x = \frac{1}{3},\,x = 1\end{align*}\]So, the function will stop changing at \(x = \frac{1}{3}\) and \(x = 1\).