The function `xsqrt(1-x^(2)),(x gt0)` has

Options

A. a local maxima
B. a local minima
C. neigher a local maxima nor a local minima
D. none of the above

Answer

A

Explanation

let `f(x)=xsqrt(1-x^(2))`
`impliesf'(x)=(1-2x^(2))/(sqrt(1-x^(2)))`
For maxima or minima, put `(dy)/(dx)=0`
`x=+-(1)/(sqrt(2))`
But xgt0, therefore we have
`x=(1)/(sqrt(2))`
Now, `f"(x)=(sqrt(1-x^(2))(-4x)-(1-2x^(2))(-x)/(sqrt(1-x^(2))))/((1-x^(2)))`
`=(2x^(3)-3x)/((1-x^(2))^(3//2))`
`impliesf"((1)/(sqrt(2)))=-ve`, maximum.