Let x = − 1 and x = 2 be the critical points of the function f ( x ) = x 3 + ax 2 + b lo g c ∣ x ∣ + 1 , x  = 0 . Let m and M respectively be the absolute minimum and the absolute maximum values of

Question

Let x = − 1 and x = 2 be the critical points of the function f ( x ) = x 3 + ax 2 + b lo g c ​ ∣ x ∣ + 1 , x  = 0 . Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval [ − 2 , − 2 1 ​ ] . Then ∣ M + m ∣ is equal to (Take lo g c ​ 2 = 0.7 ):

TestHub · Accepted Answer

​ f ( x ) = x 2 + ax 2 + b ℓ n ∣ x ∣ + 1 , x  = 0 f ′ ( x ) = 3 x 2 + 2 ax + x b ​ f ′ ( − 1 ) = 3 − 2 a − b = 0 f ′ ( − 2 ) = 12 + 4 a − 2 b ​ = 0 a = 2 − 9 ​ , b = 12 f ′ ( x ) = 3 x 2 − 9 x + x 12 ​ = x 3 ( x + 1 ) ( x + 2 ) 2 ​ ​ Max. at n = − 1 f ( x ) = x 2 − 2 9 ​ x 2 + 12 ln ∣ x ∣ + 1 ​ f ( − 1 ) = − 1 − 2 9 ​ + 1 = − 2 9 ​ M = − 4.5 ​ Min. value at x = − 2 ​ f ( − 2 ) = − 8 − 18 + 12 ℓ n 2 + 1 m = − 25 + 12 ℓ n 2 = − 16.6 ∣ M + m ∣ = 21.1 ​

Mathematics - Application of Derivative Question with Solution | TestHub

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