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Solutions of, Rolle’s Theorem & Lagrange’s Mean Value Theorem, ISC Previous Year’s Board Questions, from, 1998 to 2019, 1. Find ‘c’ of the Lagrange’s mean value theorem, if 𝑓 𝑥 = 𝑥 2 − 3𝑥 − 1, 𝑥𝜖, , −11 13, 7, , ,, , 7, , (ISC 1998), −11 13, , Solution : We have 𝑓 𝑥 = 𝑥 2 − 3𝑥 − 1 in, , ,, , 7, , We see that (i) 𝑓 𝑥 is continuous in, , 7, −11 13, 7, , ,, , , since it is a polynomial, , 7, , function., (ii) 𝑓 ′ 𝑥 = 2𝑥 − 3 which exists in, , −11 13, , ,, , 7, , 7, , . Thus all the, , conditions of Lagrange’s Mean Value Theorem are satisfied., So there must exist at least one real value c in, ′, , 𝑓 𝑐 =, Now, , 𝑓, , 13, 7, , 𝑓, , =, =, =, , 13, −11, −𝑓, 7, 7, 13, 11, − −, 7, 7, , 13 2, 7, 169, , 𝑓, , −11, 7, , =, =, =, =, , From (1), , 2𝑐 − 3 =, , 39, , 7, , ,, , 7, , such that, , …………..(1), 13, , −1, , 7, , −1, , 49, 7, 169 −273 −49, , = −, and, , −, , −3, , −11 13, , 49, 153, , 49, −11 2, , 7, 121, , +, , −3, , 33, , −11, 7, , − 1, , −1, , 49, 7, 121 +231 −49, 49, 303, 49, 153, 303, −, 49, 49, 13, −11, −, 7, 7, , −, , tapatisclasses.in
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Hence, Rolle’s Theorem is verified., 7. Verify Lagrange’s Mean Value Theorem for the function 𝑓 𝑥 = 3𝑥 2 − 5𝑥 + 1 defined, in the interval [2,5]., (ISC 2007), Solution :, 𝑓 𝑥 = 3𝑥 2 − 5𝑥 + 1 in [2,5], (i), Since 𝑓 𝑥 is a polynomial, it is a continuous function in [2,5], (ii) 𝑓 ′ 𝑥 = 6𝑥 − 5 exists in (2,5) ., Thus the conditions of Lagrange’s Mean value Theorem is satisfied., To verify Mean value Theorem, we have to show that there exists at-least one, point 𝑐 in the open interval (2,5) , such that, 𝑓′ 𝑐 =, , 𝑓 5 −𝑓 2, 5−2, , Now 𝑓 5 = 3 × 52 − 5 × 5 + 1 = 51, 𝑓 2 = 3 × 22 − 5 × 2 + 1 = 3, ∴ 6𝑐 − 5 =, , 51 −3, 3, , 6𝑐 − 5 = 16, 𝑐=, , , 21, , 𝑐=, , 6, 7, 2, , =3, , 1, 2, , 𝜖 2,5 ., , Hence Mean Value Theorem is verified., 8. Use Lagrange’s mean Value Theorem to determine a point P on the curve 𝑦 = 𝑥 − 2, defined in the interval [2,3] where the tangent is parallel to the chord joining the end, points on the curve., (ISC 2008), Solution :, We have 𝑦 = 𝑓 𝑥 = 𝑥 − 2 in [2,3]., (i), 𝑓 𝑥 is continuous in [2,3]., (ii), , 𝑓′ 𝑥 =, , 1, 2 𝑥−2, , exists in (2,3) ., , Thus all the conditions of Lagrange’s Mean Value Theorem is satisfied., ∴ by the Mean Value Theorem , we have 𝑓 ′ 𝑐 =, , 𝑓 3 −𝑓 2, 3−2, , tapatisclasses.in
