Question 1180906
 {{{F(x) =1/(x^2 +18x+84)}}}



a) give the domain of f in interval novation

domain:  {{{r}}} all real numbers  (no matter what is {{{x}}}, denominator will not be equal to zero

in interval novation:({{{-infinity}}},{{{infinity}}})


b) find the critical numbers

critical points are points where the function is defined and its derivative is zero or undefined

so, find derivative

{{{F}}}'{{{(x)}}} ={{{(d/dx)(1/(x^2 +18x+84))}}}

{{{F}}}'{{{(x)}}} = {{{-(2x + 18)/(x^2 + 18 x + 84)^2}}}

{{{F}}}'{{{(x)}}}={{{ -(2(x + 9))/(x^2 + 18 x + 84)^2}}}

equal it to zero
{{{0= -(2(x + 9))/(x^2 + 18 x + 84)^2}}}...will be zero if numerator equal to zero
{{{0= -2(x + 9)}}}
{{{0=x + 9}}}
{{{x =-9}}}



c) determine intervals where f is increasing and decreasing
Use the derivative test to determine

if {{{F}}}'{{{(x)>0}}} f is increasing
if {{{F}}}'{{{(x)<0}}} f is decreasing

in your case
Increasing:({{{-infinity}}},{{{-9}}})
Decreasing:({{{-9}}},{{{infinity }}}


a) relative maxima
{{{F(x) =1/(x^2 +18x+84)}}}, substitute {{{x =-9}}}
{{{F(x) =1/((-9)^2 +18(-9)+84)}}}
{{{F(x) =1/(81 -162+84)}}}
{{{F(x) =1/3}}}

max at ({{{-9}}},{{{1/3}}})

b) relative minima

no global minima found


{{{ graph( 600, 600, -20, 5, -5, 5, 1/(x^2 +18x+84)) }}}