The equation of a curve is given by y=f(x) wheref′(x) is a continuous function. The tangent at points(1, f(1)) , (2,f(2)) and (3,f(3)) make angles π6,π3and π4 respectively with positive x-axis. Then∫23 f′(x)f′′(x)dx+∫13 f′′(x)dx is

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−13

answer is B.

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Detailed Solution

∫23 f′(x)f′′(x)dx+∫13 f′′(x)dx=f′(x)2223+f′(x)13=12f′(3)2−f′(2)2+f′(3)−f′(1)=12tan π42−tan π32+tan π4−tan π6=12[1−3]+1−13=−13

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