Let `f` be a function with continuous second derivative and `f(0)=f^(prime)(0)=0.` Determine a function `g` by `g(x)={(f(x))/x ,x!=0 0,x=0` Then which of the following statements is correct? `g` has a continuous first derivative `g` has a first derivative `g` is continuous but `g` fails to have a derivative `g` has a first derivative but the first derivative is not continuous

Options

A. g has a continuous first derivative
B. g has a first derivative
C. g is continuous but g fails to have a derivative
D. g has a first derivative but the first derivative is not continuous

Answer

A, B

Explanation

`g(x)={{:((f(x))/(x)",",xne0),(0",",x=0):}`
`underset(xrarr0)(lim)(f(x))/(x)((0)/(0)"form")`
`=underset(xrarr0)(lim)(f'(x))/(1)`
`=f'(0)=0`
Thus g(x) is continuous at x = 0
`g'(0^(+))=underset(hrarr0)(lim)((f(h))/(h)-g(0))/(h)`
`=underset(hrarr0)(lim)(f''(h))/(2)`
`=(f''(0))/(2)`
Similarly
`g'(0^(-))=(f''(0))/(2)`