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Consider a function f(x) which satisfies f^ (x)= ^2 x+ _ - 4 ^ 4 f(x) d x . Also f ( 4 )= - 4 . If the value of 8+ ^2 _ - 4 ^ 4 f(x) d x=m , then find the value of m^2 .

Correct answer

16

Step-by-step solution

f^ (x)= ^2 x+K where K= _ - 4 ^ 4 f(x) d x aligned & f(x)= x-x+K x+C & f ( 4 )=1- 4 + K 4 +C= - 4 aligned C+1= -K 4 f(x)= x-x+K x- K 4 -1 Now, K= _ - 4 ^ 4 ( x-x+K x _ odd function - K 4 -1) d x= - 2 - K ^2 8 Hence, K= -4 8+ ^2 8+ ^2 _ - 4 ^ 4 f(x) d x=-4 m m^2=16

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