Define the function $f$ 

$f\left(x\right)\=x \cos \left(x\right)$$\stackrel{\text{assign as function}}{\to }$$f$

Graph $f\,f\prime \,f″$ 

Obtain the critical numbers $c_1$ and $c_2$ by solving the equation $f\prime \left(x\right)\=0$ 

Loading Student:-Calculus1

$f\prime \left(x\right)\=0$

$\cos \left(x\right)-x\sin \left(x\right)\=0$

$\stackrel{\text{roots}}{\to }$

$\left[0.8603335890\,3.425618459\right]$

$\stackrel{\text{assign to a name}}{\to }$

$c$

Apply the Second-Derivative test

$f″\left(c_1\right)$ = $-2.077216671$$$

$f\left(c_1\right)$ = $0.5610963382$$$

$f″\left(c_2\right)$ = $3.848816238$$$

$f\left(c_2\right)$ = $-3.288371396$$$

Obtain candidates for inflection points by solving the equation $f″\left(x\right)\=0$ 

$f″\left(x\right)\=0$

$-2\sin \left(x\right)-x\cos \left(x\right)\=0$

$\stackrel{\text{roots}}{\to }$

$\left[2.288929728\,5.086985094\right]$

$\stackrel{\text{assign to a name}}{\to }$

$p$

The individual candidates can now be referenced by typing $p_1$ and $p_2$.

Find the $x$-intercepts by solving the equation $f\left(x\right)\=0$ 

$f\left(x\right)\=0$

$x\cos \left(x\right)\=0$

$\stackrel{\text{roots}}{\to }$

$\left[0\,\frac{1}{2}\mathrm{\π}\,\frac{3}{2}\mathrm{\π}\right]$

Loading Student:-Calculus1

Applicable Commands

$\mathrm{CriticalPoints}\left(f\left(x\right)\,x\=0..2 \mathrm{\π}\,\mathrm{numeric}\right)$ = $\left[0.8603335890\,3.425618459\right]$$$

$\mathrm{ExtremePoints}\left(f\left(x\right)\,x\=0..2 \mathrm{\π}\,\mathrm{numeric}\right)$ = $\left[0.\,0.8603335890\,3.425618459\,6.283185308\right]$$$

$\mathrm{InflectionPoints}\left(f\left(x\right)\,x\=0..2 \mathrm{\π}\,\mathrm{numeric}\right)$ = $\left[2.288929728\,5.086985094\right]$$$

$\mathrm{Roots}\left(f\left(x\right)\,x\=0..2 \mathrm{\pi }\,\mathrm{numeric}\right)$ = $\left[0.\,1.570796327\,4.712388980\right]$$$

$\mathrm{Roots}\left(f\left(x\right)\,x\=0..2 \mathrm{\π}\right)$ = $\left[0\,\frac{1}{2}\mathrm{\π}\,\frac{3}{2}\mathrm{\π}\right]$$$