1) An inflection point is where the curve of the graph goes from concave down to up or vice versa. Point of inflection occur at f"=0

2) Relationship of stationary point and critical point.

We say \(\displaystyle{x}_{{0}}\) is a stationary point of a function if f(x) and f'(x) exist and is equal to \(\displaystyle{f}'{\left({x}_{{0}}\right)}={0}.\)

And, \(\displaystyle{x}_{{0}}\) is a critical point of a function of f(x) if \(\displaystyle{f{{\left({x}_{{0}}\right)}}}\) exists and either \(\displaystyle{f}'{\left({x}_{{0}}\right)}\) does not exist (i.e. function is not differentiable at or \(\displaystyle{f}'{\left({x}_{{0}}\right)}={0}.\)

All stationary points are critical points but not all critical points are stationary points.

3) Point \(\displaystyle{x}_{{0}}\) is a critical point of a function of f(x) if \(\displaystyle{f{{\left({x}_{{0}}\right)}}}\) exists and either \(\displaystyle{f}'{\left({x}_{{0}}\right)}\) does not exist (i.e. function is not differentiable at \(\displaystyle{x}_{{0}}\) or \(\displaystyle{f}'{\left({x}_{{0}}\right)}={0}.\)

The first derivative test is a method of analyzing functions using their first derivatives in order to find their extremum point.

We take the derivative of the function,

\(\displaystyle{f}'{\left(\theta\right)}=\frac{{d}}{{{d}\theta}}\cdot{\left({\sin{\theta}}{{\cos}^{{2}}\theta}-\frac{{{\cos{\theta}}}}{\theta}+{1}\right)}\)

\(\displaystyle=\frac{{d}}{{{d}\theta}}\cdot{\left({\sin{\theta}}{{\cos}^{{2}}\theta}\right)}-\frac{{d}}{{{d}\theta}}{\left(\frac{{{\cos{\theta}}}}{\theta}\right)}+\frac{{d}}{{{d}\theta}}{\left({1}\right)}\)

\(\displaystyle={{\cos}^{{3}}\theta}-\frac{{-\theta{\cos{{e}}}{c}^{{2}}\theta-{\cot{\theta}}}}{\theta^{{2}}}-{\sin{{2}}}\theta{\sin{\theta}}\)

Now, substitute \(\displaystyle\theta={4}\)

\(\displaystyle{f}'{\left({0}\right)}=-{0.27927}-{\left(-{0.49047}\right)}-{\left(-{0.74875}\right)}={0.959951}\stackrel{\sim}{=}{0.96}\)

2) Relationship of stationary point and critical point.

We say \(\displaystyle{x}_{{0}}\) is a stationary point of a function if f(x) and f'(x) exist and is equal to \(\displaystyle{f}'{\left({x}_{{0}}\right)}={0}.\)

And, \(\displaystyle{x}_{{0}}\) is a critical point of a function of f(x) if \(\displaystyle{f{{\left({x}_{{0}}\right)}}}\) exists and either \(\displaystyle{f}'{\left({x}_{{0}}\right)}\) does not exist (i.e. function is not differentiable at or \(\displaystyle{f}'{\left({x}_{{0}}\right)}={0}.\)

All stationary points are critical points but not all critical points are stationary points.

3) Point \(\displaystyle{x}_{{0}}\) is a critical point of a function of f(x) if \(\displaystyle{f{{\left({x}_{{0}}\right)}}}\) exists and either \(\displaystyle{f}'{\left({x}_{{0}}\right)}\) does not exist (i.e. function is not differentiable at \(\displaystyle{x}_{{0}}\) or \(\displaystyle{f}'{\left({x}_{{0}}\right)}={0}.\)

The first derivative test is a method of analyzing functions using their first derivatives in order to find their extremum point.

We take the derivative of the function,

\(\displaystyle{f}'{\left(\theta\right)}=\frac{{d}}{{{d}\theta}}\cdot{\left({\sin{\theta}}{{\cos}^{{2}}\theta}-\frac{{{\cos{\theta}}}}{\theta}+{1}\right)}\)

\(\displaystyle=\frac{{d}}{{{d}\theta}}\cdot{\left({\sin{\theta}}{{\cos}^{{2}}\theta}\right)}-\frac{{d}}{{{d}\theta}}{\left(\frac{{{\cos{\theta}}}}{\theta}\right)}+\frac{{d}}{{{d}\theta}}{\left({1}\right)}\)

\(\displaystyle={{\cos}^{{3}}\theta}-\frac{{-\theta{\cos{{e}}}{c}^{{2}}\theta-{\cot{\theta}}}}{\theta^{{2}}}-{\sin{{2}}}\theta{\sin{\theta}}\)

Now, substitute \(\displaystyle\theta={4}\)

\(\displaystyle{f}'{\left({0}\right)}=-{0.27927}-{\left(-{0.49047}\right)}-{\left(-{0.74875}\right)}={0.959951}\stackrel{\sim}{=}{0.96}\)