First of all we need to review the concept of concavity. So, this is related to the second derivative. If we want to think about f double prime, then we need to think about how f prime changes, how the slopes of the tangent lines change.
So:
1) On intervals where \(f''>0\), the function is concave up (Depicted in bold purple in Figure 1)
2) On intervals where \(f''<0\), the function is concave down (Depicted in bold green in Figure 1)
Points where the graph of a function changes from concave up to concave down, or vice versa, are called inflection points.
Suppose we have a function whose graph is shown below. Therefore we have:
(a) The graph of f has a point of inflection somewhere between x = -1 and x= 3
This is true. As you can see from the Figure 2 the inflection point is pointed out in green. In this point the function changes from concave down to concave up.
(b) f'(-1) = 0
This is true. In \(x=-1\) there's a maximum point. In this point the slope of the tangent line is in fact zero, that is, the function has an horizontal line as shown in Figure 3 (the line in green).
(c) this is wrong
This is false because we have demonstrated that the previous statement are true.
(d) The graph of f has a horizontal tangent line at x = 3
This is true. As in case (b) the function has an horizontal line as shown in Figure 3 (the line in orange) because in \(x=3\) there is a minimum point.
(e) The graph of f intersects both axes
This is true according to Bolzano's Theorem. Apaticular case of the the Intermediate Value Theorem is the Bolzano's theorem. Suppose that \(f(x)\) is a continuous function on a closed interval \([a,b]\) and takes the values of the opposite sign at the extremes, and there is at least one \(c \in (a,b) \ such \ that \ f(c)=0\)
