Question 1184172


Rolle’s Theorem  -  Let {{{f(x)}}} be continuous on the closed interval [{{{a}}}, {{{b}}}] and differentiable on the interval [{{{a}}}, {{{b}}}]. 

If {{{f( a) =f(b)}}} then there is at least one number {{{c}}} in [{{{a}}}, {{{b}}}] such that {{{f}}}' {{{(c)=0}}}. 


{{{f(x) = 4x^4 }}} in the interval [{{{-2}}},{{{ 2}}}] 


check: 
if continuous on  [{{{-2}}},{{{ 2}}}] 
if differentiable on  [{{{-2}}},{{{ 2}}}] 
if {{{f(-2) =f(2)}}}


The function {{{f(x) =4x^4}}} is continuous and differentiable on  [{{{-2}}},{{{ 2}}}].

{{{f(-2) = 4(-2)^4=64}}}
{{{f(2) = 4(2)^4=64}}} =>{{{f(-2) =f(2)}}}

=> role's theorem can be applied

{{{f}}}'{{{(x) = 16x^3}}}

{{{f}}}'{{{ (c)=0}}} gives {{{16x^3=0}}}  => {{{c=0}}}


and

{{{g(x) = pi*x}}}

is continuous and differentiable on [{{{-2}}}, {{{2}}}]
{{{g(-2) = pi*(-2)=-2pi}}}
{{{g(2) = pi*(2)=2pi}}}

{{{g}}}'{{{(x) =pi}}}

=> since {{{g(-2) <>g(2)}}} , role's theorem cannot be applied