Question 63932
determine the critical numbers and the critical points for each of the following functions:show all work


A)f(x)= x^3 + 3x^2 -33x -35
f'(x)=3x^2+6x-33
{{{0=3(x^2+2x-11)}}}
{{{0/3=3(x^2+2x-11)/3}}}
{{{0=x^2+2x-11}}}  We can't solve this by factoring, so use the quadratic formula:  {{{x=(-b+-sqrt(b^2-4ac))/2a}}}
a=1, b=2, c=-11
{{{x=(-2+-sqrt((2)^2-4(1)(-11)))/(2(1))}}}
{{{x=(-2+-sqrt(4+44))/2}}}
{{{x=(-2+-sqrt(48))/2}}}
{{{x=(-2+-sqrt(16)*sqrt(3))/2}}}
{{{x=(-2-4*sqrt(3))/2}}} and {{{x=-2+4*sqrt(3))/2}}}
{{{x=-1-2*sqrt(3)}}} and {{{x=1+2*sqrt(3)}}}
:
B) g(x)=x^3 - 36x
g'(x)=3x^2-36
{{{0=3x^2-36}}}
{{{0+36=3x^2-36+36}}}
{{{36=3x^2}}}
{{{36/3=3x^2/3}}}
{{{12=x^2}}}
+\-{{{sqrt(12)=sqrt(x^2)}}}
+\-{{{sqrt(4)*sqrt(3)=x}}}
+\-{{{2*sqrt(3)=x}}}
:
C) h(x)= x^3 -3x^2 -40x
h'(x)=3x^2-6x-40  
0=3x^2-6x-40  use the quadratic formula again.
a=3, b=-6, and c=-40
{{{x=(-(-6)+-sqrt((-6)^2-4(3)(-40)))/(2(3))}}}
{{{x=(6+-sqrt(36+480))/6}}}
{{{x=(6+-sqrt(516))/6}}}
{{{x=(6+-sqrt(4)*sqrt(129))/6}}}
{{{x=(6-2*sqrt(129))/6}}} and {{{x=(6+2*sqrt(129))/6}}}
{{{x=2(3-sqrt(129))/(2*3)}}} and {{{x=2(3+sqrt(129))/(2*3)}}}
{{{x=(3-sqrt(129))/3}}} and {{{x=(3+sqrt(129))/3}}}
:
Happy Calculating!!!