The value of the parameter a such that the area bounded by y = a 2 x 2 + a x + 1 , the positive coordinate axes and the line x = 1 attains its least value, is equal to

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A = ∫ 0 1 ​ ( a 2 x 2 + a x + 1 ) d x = 3 a 2 ​ + 2 a ​ + 1. d a d A ​ = 3 2 a ​ + 2 1 ​ = 0 gives a = − 4 3 ​ , and d a 2 d 2 A ​ = 3 2 ​ > 0 , so this is the minimum.

Mathematics - Area Under Curves Question with Solution | TestHub

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