If (a,0), agt 0, is the point where the curve y=sin2x−3sinx cuts the x-axis first, A is the area bounded by this part of the curve, the origin and the positive x-axis. Then
If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x=a,x=b,y=f(x)andy=f(c)(where c∈(a,b)) is minimum when c=2a+b.
Proof : A=a∫c(f(c)−f(x))dx+c∫b(f(c))dx
Differentiating w.r.t. c, we get
For maxima and minima , dcdA=0
Also for c<2a+b,dcdA<0 and for c>2a+b,dcdA>0
Hence, A is minimum when c=2a+b.
If the area enclosed by f(x)=sinx+cosx,y=a between two consecutive points of extremum is minimum, then the value of a is