# Relationships between first and second derivatives

## Homework Statement

Figure 4.46 shows the second derivative of h(x) for -2 $$\leq$$ x $$\leq$$ 1

a) Explain why h'(x) is never negative on this interval.
b) Explain why h(x) has a global maximum at x=1
c) Sketch a possible graph of h(x) for this interval.

I realize this is probably a fairly simple question, however it's just making my head hurt.

D H
Staff Emeritus
Are you sure you have given us all of the data? Perhaps the value of h'(x) at x=-1?

sorry, don't know how i could have missed that

h'(-1)=0
h(-1)=2

D H
Staff Emeritus
Much better.

The relation between the first and second derivative is that the second derivative of some function is the first derivative of the first derivative of that function.

So, to avoid this confusion, for a while let's denote h'(x) as g(x). Then g'(x)=h''(x). Now, if I showed you that graph in the first plot and labeled it as g'(x) rather than h''(x) and asked you to explain why g(x) is never negative on the interval in question, could you do that?

HallsofIvy
For $x\ge -1$, the graph shows that h"(x) is positive. That means h'(x) is increasing. And since h'(-1)= 0 ....