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If functions f(x) and g(x) are continuous on the interval [a, b] and g(x) retain the same sign on [a, b] then there is c ∈ (a , b) such that . This is known as Mean-Value Theorem. This result can be used to estimate some definite integrals. Other results which can be used for estimation are
(i)  If f increases and has a concave graph in the interval [a, b] then

(ii) If f increases and has a convex graph in the interval [a, b] then  

Q. 
Using Mean-Value Theorem, the best upper bound of​ 
  • a)
    (π/4) sin 1  
  • b)
    π sin 1  
  • c)
    (π/2) sin 1  
  • d)
    (π/4) sin (1/2)  
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If functions f(x) and g(x) are continuous on the interval [a, b] and g...
Consider Lagrange's mean value theorem for f(x) and g(x) in (b,a).

∴f′(x)=b−af(b)−f(a)​ and g′(x)=b−ag(b)−g(a)​ have atleast one real solution each.

Hence, a linear combination of these equations should also have atleast one real solution.

∴f(a)g′(x)−g(a)f′(x)=f(a)(b−ag(b)−g(a)​)−g(a)(b−af(b)−f(a)​)

∴(b−a)(f(a)g′(x)−g(a)f′(x))=f(a)g(b)−f(a)g(a)−g(a)f(b)+g(a)f(a)

∴f(a)g(a)​f(b)g(b)​=(b−a)f(a)g(a)​f′(x)g′(x)​

Hence, the above equation has atleast one root.
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If functions f(x) and g(x) are continuous on the interval [a, b] and g(x) retain the same sign on [a, b] then there is c (a , b) such that. This is known as Mean-Value Theorem. This result can be used to estimate some definite integrals. Other results which can be used for estimation are(i) If f increases and has a concave graph in the interval [a, b] then(ii) If f increases and has a convex graph in the interval [a, b] then Q.Using Mean-Value Theorem, the best upper bound of a)(/4) sin 1 b) sin 1 c)(/2) sin 1 d)(/4) sin (1/2) Correct answer is option 'A'. Can you explain this answer?
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If functions f(x) and g(x) are continuous on the interval [a, b] and g(x) retain the same sign on [a, b] then there is c (a , b) such that. This is known as Mean-Value Theorem. This result can be used to estimate some definite integrals. Other results which can be used for estimation are(i) If f increases and has a concave graph in the interval [a, b] then(ii) If f increases and has a convex graph in the interval [a, b] then Q.Using Mean-Value Theorem, the best upper bound of a)(/4) sin 1 b) sin 1 c)(/2) sin 1 d)(/4) sin (1/2) Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If functions f(x) and g(x) are continuous on the interval [a, b] and g(x) retain the same sign on [a, b] then there is c (a , b) such that. This is known as Mean-Value Theorem. This result can be used to estimate some definite integrals. Other results which can be used for estimation are(i) If f increases and has a concave graph in the interval [a, b] then(ii) If f increases and has a convex graph in the interval [a, b] then Q.Using Mean-Value Theorem, the best upper bound of a)(/4) sin 1 b) sin 1 c)(/2) sin 1 d)(/4) sin (1/2) Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If functions f(x) and g(x) are continuous on the interval [a, b] and g(x) retain the same sign on [a, b] then there is c (a , b) such that. This is known as Mean-Value Theorem. This result can be used to estimate some definite integrals. Other results which can be used for estimation are(i) If f increases and has a concave graph in the interval [a, b] then(ii) If f increases and has a convex graph in the interval [a, b] then Q.Using Mean-Value Theorem, the best upper bound of a)(/4) sin 1 b) sin 1 c)(/2) sin 1 d)(/4) sin (1/2) Correct answer is option 'A'. Can you explain this answer?.
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