Which one of the following statement is correct in respect of the function f(x)=`x^(3) sin x` ?

Options

A. It has local maximum at x = 0
B. It has local minimum at x=0
C. It has neither maximum nor minimum at x =0
D. It has maximum value 1

Answer

A

Explanation

`f(x)=x^(3) sin x`
`f'(x)=3x^(2) sin x+x^(3) cos x`
f'(x)=0
`rarr 3x^(2) sin x+x^(3) cos x=0`
`rarr 3x^(2) sinx +x^(3) cos x =0`
`rarr x=0,3 sinx + x cos x =0`
`put x =0 in (1)
`f''(x)=6x sin x + 3x^(2) cos x+x^(3)(-sinx )`
`f''(0)=0 `
so lf(x) has min at x=0