Hello, rawkstar!

This is graph shows the derivative of a a continuous function $\displaystyle f$ for [0,4].

D) If $\displaystyle f(1)=1$, graph $\displaystyle f(x)$ on [0,4].

I believe the graph looks something like this.

Code:

| P
| o *
| * :* * :
1 + o : * * :
| * : : * :
| * : : : :
| : : : :
| * : : : :
- - + - - - + - - - + - - - + - - - + - -
| 1 2 3 4

At $\displaystyle x = 0$, the slope is about $\displaystyle m = 2\tfrac{1}{2}$

The graph passes through (1,1) and when it reaches $\displaystyle x = 2$ (point $\displaystyle P$)

. . the slope has decreased to $\displaystyle m = 2.$

Then the slope suddenly changes to $\displaystyle m = -1.$

The slope gradually increases to $\displaystyle m = 0$ at $\displaystyle x = 3$ (rel. minimum)

. . and continues increasing to $\displaystyle x = 4.$

We are told that the function is *continuous*.

So at point $\displaystyle P$, the two "halves" of the graph join up.

Hence, there is a relative maximum at $\displaystyle x = 2.$