2.8 Rolle’s Theorem and the Mean Value Theorem (MVT)I. Rolle’s Theorem: Let f be continuous on [a,b] and diff’able on (a,b)If f(a)=f(b) then there is at least one c in (a,b) such that f’(c)=0. f(a)=f(b) a bExamples: Find the two x-intercepts of ( )23 2f x x x= - + and show that ( )' 0f x = at some point between the two intercepts.Work:Step 1:20 3 2x x= - +( )2 0f =( ) ( )0 2 1x x= - -( )1 0f =2 1x x= =( ) ( )2,0 1,0Step 2:Realize that f is continuous on [1,2] because it is a polynomial,Step 3: And f is diff’able on (1,2) because it is a polynomial. Step 4: ( )' 2 3f x x= -0 2 3x= -32x = See ( )3 3' 0 and 1, 22 2f� �= �� �� �*********************************************************************************************Let( )4 22f x x x= -, find all of the values for “c” in (-2,2) such that ( )' 0f c =.Work:( )3' 4 4f x x x= -( )20 4 1x x= -( ) ( )0 4 1 1x x x= + -0, 1,1x = -(all values for x are in the interval (-2,2))*********************************************************************************************II. The Mean Value Theorem (MVT): If fis continuous on [ ],a band diff’able on ( ),a bthen there exists at least one cin ( ),a bsuch that ( )( ) ( )'f b f af cb a-=-. (represents the slope of the sec line) f(b) f(b)-f(a) f(a) b-a a bExample:( )45f xx= -find all ( )1, 4c �that satisfy the MVT.Work: ( ) ( ) ( ) ( )4 14 114 1 4 1f b f a f fb a- --= = =- - -Next take the derivative:( )2' 0 4 1f x x-= - �- �( )24'f xx=Then set the derivative equal to the slope of the sec line.( )' 1f x =241x=24 x=x =+- 2(-2 is not in our interval so we have to throw out the negative answer.)**********************************************************************************************Example: ( )314xf x = +find all ( )0, 2c �that satisfy the MVT.Work:( ) ( ) ( ) ( )2 03 1 212 0 2 0 2f b f a f fb a- --= = = =- - -( )23'4f x x=2314x=243x= =23x =� 21.153c= � (Toss out negative since it is not in
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