Problem+47+Ben+Z+and+Sophia+H

page 296 #47 a) Find vertical and horizontal asymptotes There are no vertical asymptotes, but there is a horizontal asymptote at y=0. When you take the limit as x approaches infinity, the f(x) decreases, approaching, but not reaching zero.
 * f(x) = √(x²+1) - x**

b) Find intervals of increase or decrease First, you must take the derivative of the function f(x) = (x²+1)^½ - x f '(x) = ½(x²+1)^-½ ∗ 2x -1 =x(x²+1)^-½ - 1 Plugging any number into this derivative will yield a negative value. Because the derivative is always negative, the graph is decreasing over the interval (negative infinity, positive infinity)

c) Find local minimums and maximums Because local mininima and maxima occur at critical values, you have to find the critical values first. f ' (x) = x(x²+1)^-½ - 1 0 = x(x²+1)^-½ - 1 1 = x(x²+1)^-½ 1/x = 1/√(x²+1) x = √(x²+1) x² = x² + 1 There are no critical values, and therefore, there are no local mins or maxes, the graph must be decreasing the entire time

d) Find intervals of concavity and inflection points To do this, you must take the second derivative f ' (x) = x(x²+1)^-½ - 1 f '' (x) = x∗(-.5(x²+1)^-3/2 ∗2x) + (x²+1)^-½ f " (x) = -x²(x²+1)^-3/2 + (x²+1)^-½ The second derivative is positive for all real numbers, so it is concave up from (negative infinity, positive infinity)

e) Based on the information from a-d, sketch the graph