 #1
brotherbobby
 403
 103
 Homework Statement:

The figure below is the graph of a function ##y = f(x)##. Answer the following questions : ##\\[10pt]##
1. Locate the points at which the ##\text{first derivative}## of ##y## with respect to ##x## is ##\text{nonzero}##.##\\[5pt]##
2. Locate the points at which the ##\text{second derivative}## of ##y## with respect to ##x## is ##\text{nonzero}##. ##\\[10pt]##
 Relevant Equations:

1. Given a function ##f(x)##, the first derivative ##f'(x)## is 0 at a point of local extremum (maxima, minima or point of inflexion, also called saddle points). ##\\[5pt]##
2. Given a function ##f(x)##, the second derivative ##f''(x)## is negative or positive at a point of maxima or minima, respectively. At a saddle point, ##f''(x)## is zero also. ##\\[10pt]##
Problem statement : The function ##y = f(x)## is given above.
Question 1 : Locate the points at which the ##\text{first derivative}## of ##y## with respect to ##x## is ##\text{nonzero}##.##\\[5pt]##
At points of extrema, like A, C and D, the derivative is zero. Hence the derivative is non zero at the remaining points : ##\boxed{B, E, F}##.
Question 2 : Locate the points at which the ##\text{second derivative}## of ##y## with respect to ##x## is ##\text{nonzero}##. ##\\[10pt]##
Now this is the more difficult bit, because no graph of the derivative ##f'(x)## is given. However, we know that at ##A## (a maxima), derivative is negative, hence nonzero##\huge{\checkmark}##. At ##C## (a minima), derivative is positive, hence nonzero##\huge{\checkmark}##. At saddle point ##D##, second derivative (and first derivative) is zero. It gets a bit tricky at points ##B, E, F##. At ##B##, ##f'(x)## is negative and (looks to) remain the same before and after. So if we assume ##f'(B)## to be locally constant, ##f''(B)## must be zero. ##\huge{\times}## The same can be said for points ##E,F##; at all these points, the function ##f(x)## is either monotically increasing or decreasing implying that the derivative is constant and that the second derivative is zero. ##\huge{\times}##
So my answer is, at points ##\boxed{A,C}## are the second derivatives of the function nonzero.
Are my answers correct? A hint would be welcome.