Question 1155973
Perform a first derivative test on the function 

{{{f(x)= x*sqrt(64-x^2) }}}

[{{{-8}}},{{{8}}}].



{{{f}}}'{{{ (x) =(d/dx)x*sqrt(64-x^2) +x((d/dx)sqrt(64-x^2))}}}

{{{f}}}' {{{(x) =1*sqrt(64-x^2) +x( -x/sqrt(64 - x^2))}}}

{{{f}}}' {{{(x) =sqrt(64-x^2)  -x^2/sqrt(64 - x^2))}}}

{{{f}}}'{{{ (x) =(sqrt(64-x^2)*sqrt(64 - x^2)  -x^2)/(sqrt(64 - x^2))^2}}}

{{{f}}}' {{{(x) =(64 - x^2  -x^2)/(64 - x^2)}}}

{{{f}}}' {{{(x) =(64 - 2x^2 )/(64 - x^2)}}}

set {{{f}}}'{{{ (x) =0}}}

{{{(64 - 2x^2 )/(64 - x^2)=0}}}


will be zero if

{{{(64 - 2x^2 )=0}}}

{{{64 =2x^2}}}

{{{x^2=32}}}

{{{x=sqrt(32)}}}

{{{x}}}=± {{{4sqrt(2)}}}  => both are in given  [{{{-8}}},{{{8}}}] 

=> extreme points are at

{{{x=4sqrt(2) }}} and  {{{x=-4sqrt(2)}}}

or

{{{x=5.7 }}} and  {{{x=-5.7}}}


use second derivate test to determine where is max and where is min

{{{f}}}''{{{(x) = -(128 x)/(64 - x^2)^2}}}


{{{f}}}''{{{(5.7) =-(128 *5.7)/(64 - (5.7)^2)^2=-0.7}}} => negative, the absolute maximum is at {{{5.7}}}


{{{f}}}'{{{(-5.7) =-(128 *-5.7)/(64 - (-5.7)^2)^2=0.7}}} => positive, the absolute minimum is at {{{-5.7}}}



{{{ graph( 600, 600, -10, 10, -40, 40, x*sqrt(64-x^2)) }}}