The graph of is rotated counterclockwise about the origin through an acute angle $\theta$. What is the largest value of $\theta$ for which the rotated graph is still the graph of a function? What about if the graph is rotated clockwise?

To answer this question we need to find the maximal slope of $y=f(x)$, which is , and the minimal slope which is .

Thus the maximal acute angle through which the graph can be rotated counterclockwise is $\theta=$ degrees.

Thus the maximal acute angle through which the graph can be rotated clockwise is $\theta=$ degrees. (Your answer should be negative to indicate the clockwise direction.)

Note that a line $y = mx+b$ makes angle $\alpha$ with the horizontal, where $\tan(\alpha) = m$.

Hints: Recall that a graph of a function is characterized by the property that every vertical line intersects the graph in at most one point. In view of this:
1. If ALL lines $y=mx+b$ of a fixed slope $m$ intersect a graph of $y=f(x)$ in at most one point, what can you say about rotating the graph of $y=f(x)$?
2. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what can you say about rotating the graph of $y=f(x)$?
3. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what does the Mean Value Theorem tell us about $f'(x)$?

You can earn partial credit on this problem.