Maths-
General
Easy

Question

The value of 'c' in Lagrange's mean value theorem for f space left parenthesis x right parenthesis equals x cubed minus 2 x squared minus x plus 4 in [0, 1] is

  1. 1/3    
  2. 1/2    
  3. 2/3    
  4. 0    

Hint:

By Lagrange's Mean Value Theorem
 Lagrange mean value theorem states that for any two points on the curve there exists a point on the curve such that the tangent drawn at this point is parallel to the secant through the two points on the curve.
f apostrophe left parenthesis c right parenthesis space equals space fraction numerator f left parenthesis b right parenthesis space minus space f left parenthesis a right parenthesis over denominator b space minus space a end fraction

The correct answer is: 1/3


     Given : f space left parenthesis x right parenthesis equals x cubed minus 2 x squared minus x plus 4 is a polynomial in the range [0, 1]
    We know that a polynomial in the range is continuous and differentiable
    rightwards double arrow space f space left parenthesis x right parenthesis equals x cubed minus 2 x squared minus x plus 4 space space i s space c o n t i n u o u s space a n d space d i f f e r e n t i a b l e space

    By Lagrange's Mean Value Theorem
     Lagrange mean value theorem states that for any two points on the curve there exists a point on the curve such that the tangent drawn at this point is parallel to the secant through the two points on the curve.
    f apostrophe left parenthesis c right parenthesis space equals space fraction numerator f left parenthesis b right parenthesis space minus space f left parenthesis a right parenthesis over denominator b space minus space a end fraction

    rightwards double arrow space f space left parenthesis 0 right parenthesis equals 4 space space space a n d space f left parenthesis 1 right parenthesis space equals space 2


f apostrophe left parenthesis c right parenthesis space equals space fraction numerator f left parenthesis b right parenthesis space minus space f left parenthesis a right parenthesis over denominator b space minus space a end fraction space equals space fraction numerator f left parenthesis 1 right parenthesis space minus space f left parenthesis 0 right parenthesis over denominator 1 space minus space 0 end fraction space equals space fraction numerator 2 minus 4 over denominator 1 end fraction space equals space minus 2 space...... space left parenthesis 1 right parenthesis

N o w space d i f f e r e n t i a t i n g space t h e space e q u a t i o n

rightwards double arrow space f apostrophe space left parenthesis x right parenthesis equals 3 x squared minus 4 x minus space 1

rightwards double arrow space f apostrophe space left parenthesis c right parenthesis equals 3 c squared minus 4 c minus space 1

F r o m space left parenthesis 1 right parenthesis

rightwards double arrow 3 c squared minus 4 c minus space 1 space equals space minus 2

rightwards double arrow 3 c squared minus 4 c space plus space 1 space equals space 0

U sin g space Q u a d r a t i c space F o r m u l a space t o space f i n d space v a l u e space o f space c

c space equals space fraction numerator negative left parenthesis negative 4 right parenthesis space plus-or-minus square root of left parenthesis negative 4 right parenthesis squared space minus space 4 space cross times 3 space cross times 1 end root over denominator 2 cross times 3 end fraction space equals space fraction numerator 4 space plus-or-minus space 2 over denominator 6 end fraction space equals space 1 space o r space 1 third

T h u s space c space equals space 1 third

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