Question

Find the critical numbers of the function.

35. \(g\left( y \right) = \frac{{y - {\bf{1}}}}{{{y^{\bf{2}}} - y + {\bf{1}}}}\)

Short answer

The critical numbers are \(y = 0\), and 2.

Step-by-step solution

Critical Numbers of a function

TheCritical numbersfor any function \(g\left( y \right)\) are obtained by putting \(g'\left( y \right) = 0\).

Differentiating the function for critical numbers:

The given function is:

\(g\left( y \right) = \frac{{y - 1}}{{{y^2} - y + 1}}\)

On differentiating with respect to\(y\)as:

\(\begin{array}{c}g'\left( y \right) = \frac{d}{{dy}}\left( {\frac{{y - 1}}{{{y^2} - y + 1}}} \right)\\ = \frac{{\left( {{y^2} - y + 1} \right)\frac{d}{{dy}}\left( {y - 1} \right) - \left( {y - 1} \right)\frac{d}{{dy}}\left( {{y^2} - y + 1} \right)}}{{{{\left( {{y^2} - y + 1} \right)}^2}}}\\ = \frac{{{y^2} - y + 1 - 2{y^2} + 3y - 1}}{{{{\left( {{y^2} - y + 1} \right)}^2}}}\\ = \frac{{y\left( {2 - y} \right)}}{{{{\left( {{y^2} - y + 1} \right)}^2}}}\end{array}\)

Solving for critical numbers:

\(\begin{array}{c}g'\left( y \right) = 0\\\frac{{y\left( {2 - y} \right)}}{{{{\left( {{y^2} - y + 1} \right)}^2}}} = 0\\y = 0,{\rm{ }}2\end{array}\)

Hence, the critical numbers obtained are \(y = 0\), and 2.