`f(x) = sin^(-1)x+x^(2)-3x + (x^(3))/(3),x in[0,1]`

Options

A. f(x) has a point of maxima
B. f(x) has a point of minima
C. f(x) is increasing
D. f(X) is decreasing

Answer

2

Explanation

Let `f(x)=sin^(-1)x+x^(2)-3x+(x^(3))/(3)`
`therefore f(X)=(1)/sqrt(1-x^(2))+2x-3+x^(2)`
Thus f(X)=0 for some x=`x_(1) in (0,1)`

`f(x)=(x)/((1-x^(2))^(3//2))+2+2xgt 0 forall x in (0,1)`
Thus x=`x_(1)` is the point of minimum
f(0)=0,f(1)=`pi//2-5//3lt0`
f(X) is global maxima `forall x in [0,1]`.Thus
`f(X)lef(X)x in [0,1]or sin ^(-1)x+x^(2)-3x+x^(3)//3le0`
or `sin^(-1)x+x^(2)le(x9-x^(2))/(3)forall x in [0,1]`