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 ?x check your work. f(x) = x(x ? 1)e on [-0.5,2]

Solution Step 1 In this problem we need to make a complete graph of f(x) = x(x 1)e x in 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) = 0r 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 xin the neighborhood of the point p. Step 2 x Consider the following function f(x) = x(x 1)e Given the interval is [ 0.5,2] First 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) -0.5 1.24 0 0 0.5 -0.152 1 0 1.5 0.168 2 0.271 Therefore, the graph of the function is shown below Step 3 Now, at the point (0.38, 0.16), the plot has the lowest value of function. This is the lowest point in the entire plot. Thus by the definition in step 1,this point is absolute minimum. Also consider the point (1,0). Here before the point, the curve is concave up. After this point the curve is concave down. So, at this point the concavity of the figure is changed. Thus by the definition in step 1, this point be the inflection point. x So, the complete graph of the function f(x) = x(x 1)e is