The correct options are
A Area(A)=Area(B) ∀ k>0
C Area(A)=4 sq. unit when k=4
D Area(B)=1 sq. unit when k=2
f(x)=−x2+2 and g(x)=x3−x2−kx+2
Intersection point of f(x) and g(x):
−x2+2=x3−x2−kx+2
⇒x3−kx=0
⇒x=0,±√k
From the graph, we can conclude that
g(x)≥f(x) for −√k≤x≤0f(x)≥g(x) for 0≤x≤√k
So,
A=0∫−√k[g(x)−f(x)] dxB=√k∫0[f(x)−g(x)] dx
Since, g(x)−f(x)=x3−kx is an odd function, so
√k∫−√k[g(x)−f(x)] dx=0⇒0∫−√k[g(x)−f(x)] dx+∫√k0[g(x)−f(x)] dx=0⇒A−0∫√k[g(x)−f(x)] dx=0⇒A−B=0∴A=B
Now,
B=√k∫0(kx−x3)dx⇒B=[kx22−x44]√k0⇒B=k24
Therefore,
When k=4
A=B=4 sq. unit
When k=2
A=B=1 sq. unit