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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 Solution

We 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 \(x = 1\).