Curve sketching? Use the guidelines of this chapter to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. f(x) = cos2?on [? 2,2] 1+x

Solution Step 1 In this problem we need to make a complete graph of f(x) = c1+x2xin the given interval. In order to sketch the complete graph, we need to find the critical points, inflection points, local maximum and local minimum if possible. First let us see the definitions: Critical point: An interior point cof the domain of a function f at which f (c) = 0or f(c)fails to exist is called a critical point of f Inflection Point: An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. Inflection points may be stationary points, but are not local maxima or local minima. A necessary condition for x to be an inflection point is f (x) = 0 Local maximum: Let f be function defined on an interval [a,b]and let pbe a point in the open interval (a,b). Then the function f has local maximum at pif f(p) f(x) for all xin the neighborhood of the point p. Local minimum: Let f be function defined on an interval [a,b]and let pbe a point in the open interval (a,b). Then the function f has local minimum at pif f(p) f(x)for all x in the neighborhood of the point p. Step 2 Consider the following function Given the interval is [2,2] First put x = 2into the function, we get Now put x = 1.5 into the function, we get Now put x = 1 into the function, we get Now put x = 0.5 into the function, we get Now put x = 0 into the function, we get Now put x = 0.5 into the function, we get Now put x = 1 into the function, we get Now put x = 1.5 into the function, we get Now put x = 2 into the function, we get Putting these values in table we get, x f(x) -2 0.2 -1.5 0 -1 -0.5 -0.5 0 0 1 0.5 0 1 -0.5 1.5 0 2 0.2 Therefore, the graph of the function is shown below Step 3 From the above graph, note that at the points (1,0.5)and (1,0.5), the function has the minimum value in the whole curve. Thus by the definition in step 1, these two points have the lowest values of f. So, these are absolute minima. And at the points (2,0.2)and (2,0.2), the value of f is maximum with respect to the neighbor points. Thus by the definition in step 1, these points will be the local maxima. Also, at the point (0,1), the function has maximum value in the whole curve. Thus by the definition in step 1, this point is absolute maximum. Note that the graph of the function f changes its concavity at the points (1.5,0), (0.5,0), (0.5,0)and (1.5,0). Thus by the definition in s tep 1, these are the inflection points. So, the complete graph of the function f(x) = cos xis 1+x2