dse 2012 doubt

[Sris]

integral  0 to 1 (x^n sin(x) dx)
(a) Does not exist
(b) Is necessarily greater than 1
(c) Is greater than 1/(n + 1)
(d) Is less than 1/(n + 1)

A rectangle has its lower left hand corner at the origin and
its upper right hand corner on the graph of f(x) = x^2+ (1/x^2). For which x
is the area of the rectangle minimized?
(a)  x = 0
(b)  x = infinity
(c)   x =(1/3)^1/4
d)    x=2^1/3

plz help...

[L]

1) if you integrate X^n from 0 to 1 you will get 1/(n+1). Now, the area of sin x from 0 to 1 is less than 1. So, the integral should be less than 1/(n+1). this is very in loose terms, but, I hope you may have got the idea

2) You just need to minimize the area. A= x*f(X)

[Sris]

thanks L