Question

Important question for 12th board .

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Answer 1

To verify :

• Lagrange's mean value theorem for the function

F(x) = x²+x-1 in the interval [0,4]

Solution:

• F(x)=x²+x-1 in [0,4]

=> Being a polynomial F(x) is continuous and also derivable in [0,4]

Hence, Differentiate F(x) we get ,

=> dy/dx = 2x+1

Here , condition for Lagrange's mean value theorem are satisfied for F(x) in [0,4]

=> c € (0,4) satisfying

• F'(c) = ( F(b) -F(a) )/b-a

=> F' (c) = F(4)-F(0)/4-0

=>2c+1 = (4²+4-1)-(-1)/4

=>2c+1 = 20/4=5

=> c = 2

So, 2 € (0,4)

Hence , Lagrange's mean value theorem verified .

Answer 2

Answer:

To verify ↦ Lagrange's mean value theorem for the function

F(x) = x²+x-1 in the interval [0,4]

Solution:

• F(x)=x²+x-1 in [0,4]

=> Being a polynomial F(x) is continuous and also derivable in [0,4]

Hence, Differentiate F(x) we get ,

=> dy/dx = 2x+1

Here , condition for Lagrange's mean value theorem are satisfied for F(x) in [0,4]

=> c € (0,4) satisfying

• F'(c) = ( F(b) -F(a) )/b-a

=> F' (c) = F(4)-F(0)/4-0

=>2c+1 = (4²+4-1)-(-1)/4

=>2c+1 = 20/4=5

=> c = 2

So, 2 € (0,4)

Hence , Lagrange's mean value theorem verified .

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