Let f:[- , ] [-4,2] be a continuous decreasing function such that f(0)=0 and , 0 . If area enclosed by f(x) and x -axis from x=- to 0 and area enclosed by f(x) and x -axis from x=0 to x= are 1 sq . units and 3 sq . units respectively, then

Question

Let f:[- , ] [-4,2] be a continuous decreasing function such that f(0)=0 and , >0 . If area enclosed by f(x) and x -axis from x=- to 0 and area enclosed by f(x) and x -axis from x=0 to x= are 1 sq . units and 3 sq . units respectively, then area enclosed by f⁻¹(x) and x -axis from x=-4 to x=2 is (p+q +r ) . Find the value of (p+q+r).

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Correct answer: 2

aligned _ - ^0 f(x) d x+ ₂^0 f⁻¹(x) d x & =2 ₀^2 f⁻¹(x) d x & =1-2 ₀^ f(x) d x+ ₀⁻⁴ f⁻¹(x) d x & =-4 ₀⁻⁴ f⁻¹(x) d x & =-4 +3 _ -4 ^0 f⁻¹(x) d x=4 -3 Required area & = | ₀^2 f⁻¹(x) d x |+ | _ -4 ^0 f⁻¹(x) d x | & =2 -1+4 -3=2 +4 -4 p +q +r p+q+r & =2+4-4=2 aligned

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