1. 1 Example 1 Sketch the graph of the function f(x) = x 3 -3x 2 -45x+47. Solution I. Intercepts The x-intercepts are the solutions to 0= f(x) = x 3 -3x 2 -45x+47. An integer root must be a factor of 47. Clearly f(1)=0, and x=1 is the only integer root. Then f(x) = (x-1)(x 2 -2x-47). By the quadratic formula, the roots of x 2 -2x-47 are Thus the x-intercepts of f are x=1, The y-intercept of f is f(0)=47. II. Asymptotes - f has no asymptotes. III. First Derivative f / (x) = 3x 2 -6x-45 = 3(x 2 -2x-15 ) = 3(x-5)(x+3). Hence f / (x)>0 and f is increasing for x 5 while f / (x)<0 and f is decreasing for –3<x<5. f has two critical points: x=-3 and x=5. By the First Derivative Test, x=-3 is a local maximum and x=5 is a local minimum. We summarize the situation on a number line.

2. 2 IV. Vertical Tangents and Cusps f has neither cusps nor vertical tangents. V. Concavity and Inflection Points f // (x) = 6x-6 = 6(x-1). Thus f // (x) >0 and f is concave up for x>1 while f // (x) <0 and f is concave down for x<1. At x=1 the sign of f changes from negative to positive and x=1 is an inflection point. This situation is depicted on a number line. f / (x) = 3x 2 -6x- 45

3. 3 VI. Sketch of the Graph We summarize our conclusions and sketch the graph of f. x-intercepts: x=1, y-intercept: y=47 Asymptotes: none Increasing on (- ,-3)  (5,  ) and decreasing on (-3,5) Local max at (-3,128), local min at (5,-128) No vertical tangents or cusps Concave down on (- ,1) and concave up on (1,  ) Inflection point at (1,0) f(x) = x 3 -3x 2 -45x+47