- #1

brotherbobby

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- 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{non-zero}##.##\\[5pt]##

2. Locate the points at which the ##\text{second derivative}## of ##y## with respect to ##x## is ##\text{non-zero}##. ##\\[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]##

**The function ##y = f(x)## is given above.
**

Problem statement :

Problem statement :

**Question 1 :**Locate the points at which the ##\text{first derivative}## of ##y## with respect to ##x## is ##\text{non-zero}##.##\\[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{non-zero}##. ##\\[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 non-zero##\huge{\checkmark}##. At ##C## (a minima), derivative is positive, hence non-zero##\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 non-zero.

**Are my answers correct? A hint would be welcome**.