Unformatted text preview: 1996
A [14 . L This problem deals with functions defined by f(x) = x + b sin x, where b is a positive constant and
—Zrt S x S 2a. (:1) Sketch the graphs of two of these functions. y = x + sin x and y = x + 3 sin x, as indicated
below. Note: The axes for these two graphs are provided in the pink test booklet only. (b) Find the xcoordinates of all points, "21: S x S 211:, where the line y = x + b is tangent to the
graph of f(x) = x + b sin 1. (1:) Are the points of tangency described in part (b) relative maximum points of f? Why? (d) For all values of b > 0, show that all. inflection points of the graph of f lie on the line 3: a x. (a) y=x+sinx y=x+3s1nx (b) y= —x+b =ey' =1 andf(x)= —x+bsinx mf’(x)=1+bcosx
At the point of tangency, the ycoordinates are equal and the derivatives are equal x+b=x+bsinx=a=>sinx=1 machTE— or _§2£ 1': 311:
1=1+bcosx=z=>cosx=0=ax=i§ orxee?
The graph of the line )1: x+b is tangent to the graph of ﬂat) at x=~£ and x= ~22: @ (c) No the points of tangericy are not relative maximum points of f because the slope
(derivative) off at the points of tangency is equal to one, but at reiative maximum points the derivative must equal zero or be nonexistent. @ (d) f(x) =x + bsinx = f'(x) = 1 + bcosx => f”(x) = mbsinx
At all inﬂection points of f, f”(x) =0 which implies that sinx «1 0 (since b > 0) which
implies that f(x) =x. Therefore, inflection points lie on the line y =x. @ 1996 AB4(BC4) ...
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