Answer the questions below:

MATHEMATICS · Subjective · A Level · Mathematics · Differentiation · Differentiation

Problem

Answer the questions below: (i) State precisely the conditions under which a point

c

in the domain of a differentiable function

F

can be identified as a point of local maximum or local minimum on the basis of the behaviour of

F'(x)

in the neighbourhood of

c

. (ii) Give an explicit example of a differentiable function

F

and a point

c

such that

F'(c)=0

but

c

is neither a point of local maximum nor a point of local minimum. (iii) Justify, using the sign of

F'(x)

on either side of

c

, why the point given in part (ii) fails to be an extremum.