Use the given graph $f$ over the interval (0, 7) to find the following: 
(a) The open intervals on which $f$ is increasing.
I answered $(0, 1), (3, 5), (5,7)$
(b) The open intervals on which $f$ is concave upward.
I answered $(1, 4)$
(c) The open intervals on which $f$ is concave downward.
I answered $(0, 1)$
(d) The coordinates of the points of inflection.
I answered:
$(1,3)$ smallest $x$-value
$(3,1)$
$(5,4)$ largest $x$-value
WebAssign said that I got all of these wrong. I am trying to understand why. Please help!
 A: For a, it is increasing on $(3,7)$.  True, it has zero derivative at $5$, but $f(5) \gt f(x)$ for all $x$ close to and less than $5$.  For b, it is clearly concave downward at $1$.  It looks concave upward starting at $2$ for an interval $(2,4)$ and again on $(5,7)$ to me.  Then for c, concave downward is $(0,2)$ and $(4,5)$.  For d, the points of inflection are the changes in concavity, so $(2,2), (4,3), (5,4)$
A: For question a, think of what it means to be increasing. When you take the derivative, and you find critical values, you have to test points around those critical values, to determine if it is a minimum or maximum. Take the function $f(x)=x^3$. The derivative is$f(x)=3x^2$, and the critical value is $x=0$. The test intervals to find the relative extrema, are $(-\infty,0)$ and $(0, \infty)$. If you take test values, you'll see that the derivative is positive on the two intervals, meaning that that tangent lines drawn at any point on the graph will have a positive slope; furthermore, implying that the function is increasing on $(-\infty, \infty)$ So there is relationship between the original function, and it's derivative: the function is increasing at a particular x-value when it's derivative is positive at that x-value; and decreasing at a particular x-value when it's derivative is negative at that x-value.
The function you give is indeed increasing on $(0,1)$ But look back at your two other intervals, $(3,5)$ and $(5,7)$ and draw tangent lines to the graph. What are the slopes of the tangent lines? Positive, negative?
A: the open intervals on which f is concave upward. (Enter your answer using interval notation.) 
answer:(2,4),(5,7)
The open intervals on which f is concave downward. (Enter your answer using interval notation.) 
answer: (0,2),(4,5)
