`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3).` Then the least value of n for which `(d^(n))/(dx^(n))f(x)|underset(x=0)` is nonzero is `a.`5 `b.`6 `c.`7 `d.`8

Options

A. 5
B. 6
C. 7
D. 8

Explanation

`f(x)=e^(x)-e^(-x)-2 sin x -(2)/(3)x^(3)`
`f^(I)(x)=e^(x)+e^(-x)-2 cos x -2x^(2)`
`f^(II)(x)=e^(x)-e^(-x)+2 sin x-4x`
`f^(III)(x)=e^(x)+e^(-x)+2 cos x -4`
`f^(IV)(x)=e^(x)-e^(-x)-2 sin x`
`f^(V)(x)=e^(x)+e^(-x)-2 cos x`
`f^(VI)(x)=e^(x)-e^(-x)+2 sin x`
`f^(VII)(x)=e^(x)+e^(-x)+2 cos x`
`"Clearly, "f^(VII)(0)" is nonzero."`