More graphing ?Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work. f?(?x?)? ? ln. ?X

Solution:- Step1 Given function is f(x)= x In x And the domain of given function is [0, ] The given function is not an even function and not an odd function. So, the given function is not symmetrical about x-axis and y-axis. Step2 Differentiate the given equation to find f’(x) we get, f’(x)= In(x)+1 Again differentiate f’(x) to find f”(x) we get, 1 f”(x)= x Step3 To get extreme value we have to use f’(x)=0 In(x) + 1=0 In(x)=-1 x=e = 1 1 e 1 x= e is the critical point. Step4 To find the inflection points we have to use f”(x)=0 x=0 1=0 inflection point is not possible. Step5 We have to find increasing, decreasing and concavity 1 x= e is the critical point. Therefore in between ( ,) the function f(x) is increasing and (0, ) it is decreasing. e e Let f”(x)>0 1 >0 x The function is concave upwards if x>0. Step6 Extreme values and inflection points. The inflection points are at 1 f”(x)= x f”( )= 1=e>0 e e 1 Therefore at x= e f(x) has minimum value . Step7 Asymptotes and end behavior Let f’(x)=0 if x= 1 e 1 At x= e then tangent parallel to x- axis limf(x)=lim x In x= 0 x0 x0 xm f(x)=xm x In x= Intercepts At x=0 y=0 If y=0 x= 1 and 0 Step8 Graph of given equation