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.

\(f(x)=\frac{x^{2}+x}{4-x^{2}}\)

Solution Step 1 x +x In this problem we need to make a complete graph of f(x) = 4x2in its domain or in the given interval. Since the interval is not mentioned, we take the domain. Here in the given function equate denominator to 0, we get x = ±2, therefore the domain is {x; x = / ±2} . 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.