Question

# The graph of f(x)=−10x+e5sin(x)f(x)=−10x+e5sin⁡(x) is rotated counterclockwise about the origin through an acute angle θθ. What...

The graph of

f(x)=−10x+e5sin(x)f(x)=−10x+e5sin⁡(x)

is rotated counterclockwise about the origin through an acute angle θθ. What is the largest value of θθ 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)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 θ=θ= degrees.
Thus the maximal acute angle through which the graph can be rotated clockwise is θ=θ= degrees. (Your answer should be negative to indicate the clockwise direction.)
Note that a line y=mx+by=mx+b makes angle αα with the horizontal, where tan(α)=mtan⁡(α)=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+by=mx+b of a fixed slope mm intersect a graph of y=f(x)y=f(x) in at most one point, what can you say about rotating the graph of y=f(x)y=f(x)?
2. If some line y=mx+by=mx+b intersects the graph of y=f(x)y=f(x) in two or more points, what can you say about rotating the graph of y=f(x)y=f(x)?
3. If some line y=mx+by=mx+b intersects the graph of y=f(x)y=f(x) in two or more points, what does the Mean Value Theorem tell us about f′(x)f′(x)?

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