Question
The value of 'c' in Lagrange's mean value theorem for
in [0, 2] is
- 0
- 1
- 2/3
- 3/2
Hint:
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.
The correct answer is: 2/3
Given :
is a polynomial
We know that polynomials are continuous and differentiable in their range
![rightwards double arrow space f space left parenthesis x right parenthesis equals x space left parenthesis x minus 2 right parenthesis squared 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 i n space t h e space r a n g e space left square bracket space 0 comma space 2 space right square bracket](data:image/png;base64,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)
Simplify f(x) for ease of calculation
![rightwards double arrow space f space left parenthesis x right parenthesis equals x space left parenthesis x minus 2 right parenthesis squared space
rightwards double arrow space f space left parenthesis x right parenthesis equals x space left parenthesis x squared plus 4 space minus 4 x right parenthesis
rightwards double arrow space f space left parenthesis x right parenthesis equals x cubed plus 4 x space minus 4 x squared space....... space left parenthesis 1 right parenthesis
N o w comma space s u b s t i t u t e space t h e space v a l u e s space o f space x space a s space 0 space a n d space 2
rightwards double arrow space f left parenthesis 0 right parenthesis space equals space 0
rightwards double arrow space f left parenthesis 2 right parenthesis space equals space space 0 space
B y space L a g r a n g e s space M e a n space V a l u e space T h e o r e m
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 minus a end fraction
f apostrophe left parenthesis c right parenthesis space equals space fraction numerator f left parenthesis 2 right parenthesis space minus space f left parenthesis 0 right parenthesis over denominator 2 minus 0 end fraction space equals space 0 space space........ space left parenthesis 2 right parenthesis
N o w space d i f f e r e n t i a t i n g space left parenthesis 1 right parenthesis space
f apostrophe space left parenthesis x right parenthesis equals 3 x squared plus 4 space minus 8 x
f apostrophe left parenthesis c right parenthesis space equals space 3 c squared plus 4 space minus 8 c
F r o m space left parenthesis 2 right parenthesis
3 c squared plus 4 space minus 8 c space equals space 0
N o w space b y space u sin g space q u a d r a t i c space f o r m u l a comma space w e space f i n d space t h e space r o o t s space o f space c
c space equals space fraction numerator negative left parenthesis negative 8 right parenthesis space plus-or-minus square root of left parenthesis negative 8 right parenthesis squared space minus space 4 space cross times 3 cross times 4 end root over denominator 2 cross times 3 end fraction space equals space fraction numerator 8 space plus-or-minus space 4 over denominator 6 end fraction space equals space 2 space o r space 2 over 3
T h u s space c space equals space 2 over 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)
Related Questions to study
The equation
represents
The equation
represents
The polar equation of the circle whose end points of the diameter are
and
is
The polar equation of the circle whose end points of the diameter are
and
is
The radius of the circle
is
The radius of the circle
is
The adjoining figure shows the graph of
Then –
![](data:image/png;base64,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)
Here we can see that the graph was given to us and us to take out the conclusion from that since we have the options available so I would suggest you to always start to check from the options because by the use of options we can see how easily we concluded this question.
The adjoining figure shows the graph of
Then –
![](data:image/png;base64,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)
Here we can see that the graph was given to us and us to take out the conclusion from that since we have the options available so I would suggest you to always start to check from the options because by the use of options we can see how easily we concluded this question.
Graph of y = ax2 + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –
![](data:image/png;base64,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)
Here we can see that the graph was given to us and us to take out the conclusion from that since we have the options available so I would suggest you to always start to check from the options because by the use of options we can see how easily we concluded this question.
Graph of y = ax2 + bx + c = 0 is given adjacently. What conclusions can be drawn from this graph –
![](data:image/png;base64,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)
Here we can see that the graph was given to us and us to take out the conclusion from that since we have the options available so I would suggest you to always start to check from the options because by the use of options we can see how easily we concluded this question.
For the quadratic polynomial f (x) = 4x2 – 8kx + k, the statements which hold good are
For the quadratic polynomial f (x) = 4x2 – 8kx + k, the statements which hold good are
The graph of the quadratic polynomial y = ax2 + bx + c is as shown in the figure. Then :
![](data:image/png;base64,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)
The graph of the quadratic polynomial y = ax2 + bx + c is as shown in the figure. Then :
![](data:image/png;base64,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)
The greatest possible number of points of intersections of 8 straight line and 4 circles is :
The students can make an error if they don’t know about the formula for calculating the number of points as mentioned in the hint which is as follows
The number point of intersection between two lines can be counted by finding the number of ways in which two lines can be selected out of the lot as two lines can intersect at most one point.
The number point of intersection between two circles can be counted by finding the number of ways in which two circles can be selected out of the lot multiplied by 2 as two circles can intersect at most two points.
The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.
The greatest possible number of points of intersections of 8 straight line and 4 circles is :
The students can make an error if they don’t know about the formula for calculating the number of points as mentioned in the hint which is as follows
The number point of intersection between two lines can be counted by finding the number of ways in which two lines can be selected out of the lot as two lines can intersect at most one point.
The number point of intersection between two circles can be counted by finding the number of ways in which two circles can be selected out of the lot multiplied by 2 as two circles can intersect at most two points.
The number point of intersection between two circles can be counted by finding the number of ways in which one circle and one line can be selected out of the lot multiplied by 2 as one circle and one line can intersect at most two points.
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even position ?
Here we have obtained the total number of 9 digit numbers using the given digits. While finding the number of ways to arrange the odd digits in 5 even places, we have divided the 4! by 2! because the digit 3 were occurring two times and the digit 5 were occurring 2 times. Here we can make a mistake by conserving the number of even digits 4 and the number of odd digits 5, which will result in the wrong answer.
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even position ?
Here we have obtained the total number of 9 digit numbers using the given digits. While finding the number of ways to arrange the odd digits in 5 even places, we have divided the 4! by 2! because the digit 3 were occurring two times and the digit 5 were occurring 2 times. Here we can make a mistake by conserving the number of even digits 4 and the number of odd digits 5, which will result in the wrong answer.