Lert `f:[a , b]vec` be a function such that for `c in (a , b),f^(prime)(c)=f^(c)=f^(c)=f^(i v)(c)=f^v(c)=0.` Then `f` has a local extermum at `x=cdot` `f` has neither local maximum nor minimum at `x=c` `f` is necessarily a constant function it is difficult to say whether `a)or(b)dot`

Options

A. f has local extremum at x=c
B. f has neither local maximum nor local minimum at x=c
C. f is necessarily a constant function
D. It is difficult to say whether (A) or (B)

Answer

D

Explanation

`f'(C )=f"(C )=f"'(C )=f^(iv)(C )=f^(v)(C )=0`
Now if n is the least positive integer such that `f^(n)(C )ne0`, then it is not clear whether n is even or odd. So nothing can be said whether f(x) has local extrema at or not