By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

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Finding zeros of polynomials of degree higher than 2 and other functions is usually trickier and is sometimes impossible. At the least, you can always find an approximation of any zero(s) by using a graph to zoom in closer to the point(s) where the graph crosses the x-axis, as we’ll illustrate shortly. A more basic question, though, is to determine how many zeros a given function has. In general, there is no way to answer this question without the use of calculus. 3 (a consequence of the Fundamental Theorem of Algebra) provides a clue.

1 1 2 3 4 x 5 y 34. 5 2 x ............................................................ 0 Ϫ2 In exercises 39–42, identify the given function as polynomial, rational, both or neither. Ϫ1 x 1 ............................................................ In exercises 35–38, use the vertical line test to determine whether the curve is the graph of a function. 39. f (x) = x 3 − 4x + 1 41. f (x) = x 2 + 2x − 1 x +1 x 3 + 4x − 1 x4 − 1 √ 42. f (x) = x 2 + 1 40. f (x) = ............................................................

Y y ϭ a1x ϩ a0 0-20 y-intercepts and points known as extrema are of interest. The function value f (M) is called a local maximum of the function f if f (M) ≥ f (x) for all x’s “nearby” x = M. Similarly, the function value f (m) is a local minimum of the function f if f (m) ≤ f (x) for all x’s “nearby” x = m. A local extremum is a function value that is either a local maximum or local minimum. Whenever possible, you should produce graphs that show all intercepts and extrema (the plural of extremum).