considering f(x) is a twice differentiable functionif f ' (a) =0 and f '' (a) <0 it is said to be maxima similary if f ''(a)>0 it said to be minima but sometimes this rule failed to work can u tell me why

repeatedly differentiate the function till you get the smallest value of n for which fⁿ(x)≠zero, if n is even and the  derivative is positive it is a minimum, if the derivative is negative you get a maximum,if n is odd, it is neither a maximum nor a minimum

Approved

It is called second order derivative test it is used to check whether the obtained answer is maxima or minima but sometimes we get f "(a)=0 so here it fails in this case nth order derivative test is used