Let the function, f ( x ) = { − 3 a x 2 − 2 , a 2 + b x , x 1 , b ∈ R . If the area of the region enclosed by y = f ( x ) and the line y = − 2

Question

Let the function, f ( x ) = { − 3 a x 2 − 2 , a 2 + b x , ​ x < 1 x ⩾ 1 ​ be differentiable for all x ∈ R , where a > 1 , b ∈ R . If the area of the region enclosed by y = f ( x ) and the line y = − 20 is α + β 3 ​ , α , β ∈ Z , then the value of α + β is ________

TestHub · Accepted Answer

f ( x ) is continuous and differentiable ​ at x = 1 , LHL = RHL , LHD = RHD − 3 a − 2 = a 2 + b , − 6 a = b a = 2 ; b = − 12 f ( x ) = { − 6 x 2 − 2 , 4 − 12 x , ​ x < 1 x ≥ 1 ​ ​ ​ Area = ∫ − 3 ​ 1 ​ ( − 6 x 2 − 2 + 20 ) d x + ∫ 1 2 ​ ( 4 − 12 x + 20 ) d x ] = 16 + 12 3 ​ + 6 = 22 + 12 3 ​ ∴ α + β = 34 ​

Mathematics - Continuity - Differentiability Question with Solution | TestHub

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