Question 1176786
1.

{{{p(x)=x^3 -7x^2+15x-9}}}.......factor completamente
{{{p(x)=x^3 -x^2-6x^2+6x+9x-9}}}
{{{p(x)=(x^3 -x^2)-(6x^2-6x)+(9x-9)}}}
{{{p(x)=x^2(x -1)-6x(x-1)+9(x-1)}}}
{{{p(x) = (x^2 - 6x+9) (x - 1)}}}
{{{p(x) = (x - 3)^2 (x - 1)}}}

soluciones:
{{{(x - 3)^2 (x - 1)=0}}}

si {{{(x - 3)^2 =0 }}}=>{{{(x - 3) =0}}}  =>{{{x =3}}} 
si {{{(x - 1)=0 }}}=> {{{x=1}}}


Suppose that x=c is a critical point of f (x ) then:
If f'(x)>0 to the left of x=c and f' (x )<0 to the right of x=c then x=c is a local maximum. 
If f'(x) < 0 to the left of x=c and f' (x )>0 to the right of x=c then x=c is a local minimum. 
If f (x ) is the same sign on both sides of x=c then x=c is neither a local maximum nor a local minimum.

Suponga que x = c es un punto crítico de f (x) entonces:
Si f '(x)> 0 a la izquierda de x = c y f' (x) <0 a la derecha de x = c, entonces x = c es un máximo local.
Si f '(x) <0 a la izquierda de x = c y f' (x)> 0 a la derecha de x = c, entonces x = c es un mínimo local.
Si f (x) es el mismo signo en ambos lados de x = c, entonces x = c no es ni un máximo local ni un mínimo local.

entonces, encuentre la primera derivada de la función p (x)

{{{p}}}'{{{(x)=3x^2 -14x+15}}}
{{{3x^2 -14x+15=0}}}
{{{(x - 3) (3 x - 5) = 0}}}

{{{x-3=0}}} ->{{{x=3}}}
{{{3x-5=0 }}}-> {{{x=5/3}}}

ahora encuentra

{{{p(x)=x^3 -7x^2+15x-9}}} if {{{x=3}}}
{{{p(3)=3^3 -7*3^2+15*3-9=0}}}

({{{3}}},{{{0}}})

and if {{{x=5/3}}}

{{{p(5/3)=(5/3)^3 -7*(5/3)^2+15*(5/3)-9=32/27}}}
({{{5/3}}},{{{32/27}}})

entonces usted tiene

Maximum: ({{{5/3}}},{{{32/27}}})
Minimum:({{{3}}},{{{0}}})


2.
{{{p(x)=x^4+ x^3 -4 x^2-4x}}}
{{{p(x) = x (x^3 + x^2 - 4 x - 4)}}}
{{{p(x) = x ((x^3 + x^2) - (4 x+ 4))}}}
{{{p(x) = x (x^2(x + 1) - 4(x+ 1))}}}
{{{p(x) = x ((x^2 - 4)(x+ 1))}}}
{{{p(x) = x(x - 2)(x + 2)(x + 1)}}}
soluciones:
{{{x=0}}}
{{{x=2}}}
{{{x=-2}}}
{{{x=-1}}}

{{{p}}}'{{{(x)=4x^3+ 3x^2 -8x-4}}}
{{{4x^3+ 3x^2 -8x-4=0}}}

usando la calculadora obtendrás

{{{x}}}≈{{{-1.61}}}

{{{x}}}≈{{{-0.47}}}

{{{x}}}≈{{{1.33}}}

{{{p(-1.61)=(-1.61)^4+ (-1.61)^3 -4 *(-1.61)^2-4*(-1.61)=-1.38}}}->({{{-1.61}}},{{{-1.38}}})
{{{p(-0.47)=(-0.47)^4+ (-0.47)^3 -4 *(-0.47)^2-4*(-0.47)=0.94}}}->({{{-0.47}}},{{{0.94}}})
{{{p(1.33)=(1.33)^4+ (1.33)^3 -4 *(1.33)^2-4*(1.33)=-6.91}}}->({{{1.33}}},{{{-6.91}}})

Maximum:({{{-0.47}}},{{{0.94}}})
Minimum:({{{1.33}}},{{{-6.91}}})


3.
{{{p(x)=x^5-10x^3 +9x}}}
{{{p(x)=x(x^4-10x^2 +9)}}}
{{{p(x)=x(x^4-x^2-9x^2 +9)}}}
{{{p(x)=x((x^4-x^2)-(9x^2 -9))}}}
{{{p(x)=x(x^2(x^2-1)-9(x^2 -1))}}}
{{{p(x)=x((x^2-9)(x^2 -1))}}}
{{{p(x)=x((x^2-3^2)(x^2 -1))}}}
{{{p(x)=x(x-3)(x+3)(x -1)(x+1)}}}

=> soluciones:

{{{x=0}}}
{{{x=3}}}
{{{x=-3}}}
{{{x=1}}}
{{{x=-1}}}

{{{p}}}'{{{(x)=5x^4-30x^2 +9}}}
{{{5x^4-30x^2 +9=0}}}
{{{5x^4-30x^2 +9=0}}}

usando la calculadora obtendrás

{{{x = sqrt(3 - 6/sqrt(5))}}}≈ {{{x=0.56}}}
{{{x = -sqrt(3 - 6/sqrt(5))}}}≈{{{x=-0.56}}}
{{{x = sqrt(3+6/sqrt(5))}}}≈{{{x=2.38}}}
{{{x = -sqrt(3 + 6/sqrt(5))}}}≈{{{x=-2.38}}}


{{{p(0.56)=(0.56)^5-10*(0.56)^3 +9(0.56)=3.34}}} ->({{{0.56}}},{{{3.34}}})
{{{p(-0.56)=(-0.56)^5-10*(-0.56)^3 +9(-0.56)=-3.34}}} ->({{{-0.56}}},{{{-3.34}}})
{{{p(2.38)=(2.38)^5-10*(2.38)^3 +9(2.38)=-37.03}}} ->({{{2.38}}},{{{-37.03}}})
{{{p(-2.38)=(-2.38)^5-10*(-2.38)^3 +9(-2.38)=37.03}}} ->({{{-2.38}}},{{{37.03}}})


Maximum:({{{0.56}}},{{{3.34}}}) y ({{{-0.56}}},{{{-3.34}}})
Minimum:({{{2.38}}},{{{-37.03}}}) y ({{{2.38}}},{{{-37.03}}})