Paragraph: Read the following passage and answer the questions. For every function f(x) which is twice differentiable, these will be good approximation of _a^b f(x) d x ( b-a 2 ) f(a)+f(b) . Now, if we take c= a+b 2 , then using above again

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Paragraph: Read the following passage and answer the questions. For every function f(x) which is twice differentiable, these will be good approximation of _a^b f(x) d x ( b-a 2 ) f(a)+f(b) . Now, if we take c= a+b 2 , then using above again, we get _a^b f(x) d x= _a^c f(x) d x+ _c^b f(x) d x b-a 4 f(a)+f(b)+2 f(c) and so on. We get approximation for value of _a^b f(x) d x . Question: If _ t a _a^t f(x) d x- (t-a) 2 f

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Correct answer: B. 1

aligned _ t a _a^t f(t) d t- (t-a) 2 (f(t)+f(a)) (t-a)^3 & =0 _ h 0 _a^ a+h f(t) d t- h 2 (f(a+h)+f(a)) h^3 & =0 _ h 0 f(a+h)- 1 2 (f(a+h)+f(a))- h 2 (f^ (a+h) ) 3 h^2 & =0 aligned f^ (a+h)- 1 2 f^ (a+h) _ h 0 - 1 2 f^ (a+h)- h 2 f^ (a+h) 6 h =0 _ h 0 - h 2 f^ (a+h) 6 h =0 f^ (a)=0, a R (x) must have maximum degree 1.

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