Suppose f : [ 1 , 2 ] → R is such that f ( x ) = x 3 + b x 2 + c x . If f satisfies the hypothesis of Rolle's theorem on [ 1 , 2 ] and the conclusion of Rolle's theorem holds for f on [ 1 ,

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Suppose f : [ 1 , 2 ] → R is such that f ( x ) = x 3 + b x 2 + c x . If f satisfies the hypothesis of Rolle's theorem on [ 1 , 2 ] and the conclusion of Rolle's theorem holds for f on [ 1 , 2 ] at the point 3 4 ​ , then

TestHub · Accepted Answer

f ( 1 ) = f ( 2 ) ⇒ 1 + b + c = 8 + 4 b + 2 c ⇒ 3 b + c = − 7 → ( 1 ) . Now, f ′ ( x ) = 3 x 2 + 2 b x + c ; ∴ f ′ ( 3 4 ​ ) = 0 (given) ⇒ 3. 9 16 ​ + 2 b . 3 4 ​ + c = 0 ⇒ 8 b + 3 c = − 16 → (2). From (1),(2) we get b = − 5 and c = 8 .

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