Answer
\[\begin{align}
& \left( \text{a} \right)x=0.5,\text{ }x=1,\text{ }x=1.5,\text{ }x=2,\text{ }x=2.5,\text{ }x=3,\text{ }x=3.5 \\
& \left( \text{b} \right)x\approx \frac{1}{2},\text{ }x=0.97,\text{ }x=1.5,\text{ }x=1.98,\text{ }x=2.5,\text{ }x=2.98,\text{ }x=3.5 \\
\end{align}\]
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{{{{\cos }^2}\pi x}}{{\sqrt {{x^2} + 1} }},{\text{ }}0 < x < 4 \cr
& \left( {\text{a}} \right){\text{Graph using a CAS }}\left( {{\text{shown below}}} \right) \cr
& {\text{Visually, we can notice that the critical numbers are:}} \cr
& x = 0.5,{\text{ }}x = 1,{\text{ }}x = 1.5,{\text{ }}x = 2,{\text{ }}x = 2.5,{\text{ }}x = 3,{\text{ }}x = 3.5 \cr
& \cr
& \left( {\text{b}} \right){\text{Find the }}f''\left( x \right){\text{ using a CAS }}\left( {{\text{shown below}}} \right) \cr
& {\text{Visually, we can notice that the critical numbers are:}} \cr
& x \approx \frac{1}{2},{\text{ }}x = 0.97,{\text{ }}x = 1.5,{\text{ }}x = 1.98,{\text{ }}x = 2.5,{\text{ }}x = 2.98,{\text{ }} \cr
& x = 3.5 \cr
} $$
