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Problem Questions with Answer, Solution - Exercise 7.3: Mean Value Theorem | 12th Maths : UNIT 7 : Applications of Differential Calculus

Chapter: 12th Maths : UNIT 7 : Applications of Differential Calculus

Exercise 7.3: Mean Value Theorem

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EXERCISE 7.3

1. Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.

(i) f (x) = |1/x| , x [−1,1]

(ii) f (x) = tan x, x [0,π ]

(iii) (x) = x  2 log x, x [2, 7]


2. Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the -axis for the following functions :

(i) (x) = x2  x, x [0,1]

(ii) f (x) = (x2 − 2x) / (x + 2), x  [ −1, 6]

(ii) f (x) = √x  x/3 , x [0, 9]


3. Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals :

(i) f (x) = x +1 / x, x  [ −1, 2]

(ii) f (x) = | 3x +1 |, x  [ −1, 3]


4. Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:

(i) f ( x= x3  3x + 2, x [2, 2]

(ii) f (x= ( x  2)(x  7), x [3,11]


5. Show that the value in the conclusion of the mean value theorem for

(i) f (x) = 1/x on a closed interval of positive numbers [a, b] is √(ab)

(ii) (x) = Ax2 + Bx + C on any interval [a, b] is (+b) / 2 .


6. A race car driver is kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours.


7. Suppose that for a function f ( x), f ( x)  1 for all 1  x  4 . Show that f (4)  f (1)  3 .


8. Does there exist a differentiable function  f (x) such that f (0) = −1, f (2) = 4 and f’(x) ≤ 2 for all x . Justify your answer.


9. Show that there lies a point on the curve  f (x) = x ( x + 3)e-π/2 , −3 ≤ x ≤ 0 where tangent drawn is parallel to the x -axis.


10. Using mean value theorem prove that for, a > 0, b > 0, | e−a − e−b | < | a − b | .


 

Answers:

 (1) (i) not continuous at x = 0 (ii) not continuous at x = π/2 (iii) π f ( 2) ≠ f (7)

(2) (i) 1/2 (ii) −2 + 2√2 (iii) 9/4

(3) (i) not continuous at = 0 (ii) not differentiable at x = −1/3

(4) (i) ± 2/√3  (ii) 7

(6) 320 km

(8) No. Since f ′ ( x) cannot be 2.5 at any point in ( 0, 2) .

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12th Maths : UNIT 7 : Applications of Differential Calculus : Exercise 7.3: Mean Value Theorem | Problem Questions with Answer, Solution

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12th Maths : UNIT 7 : Applications of Differential Calculus


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