The points at which the function is not precisely defined: x1=−1
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0 so we need to solve the equation: x+1xex−1=0 Solve this equation The points of intersection with the axis X:
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0: substitute x = 0 to (-1 + x*exp(x))/(1 + x). 1−1+0e0 The result: f(0)=−1 The point:
(0, -1)
Extrema of the function
In order to find the extrema, we need to solve the equation dxdf(x)=0 (the derivative equals zero), and the roots of this equation are the extrema of this function: dxdf(x)= the first derivative x+1xex+ex−(x+1)2xex−1=0 Solve this equation Solutions are not found, function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this dx2d2f(x)=0 (the second derivative equals zero), the roots of this equation will be the inflection points for the specified function graph: dx2d2f(x)= the second derivative x+1(x+2)ex−2ex+(x+1)22(xex−1)=0 Solve this equation The roots of this equation x1=−25787.3812935264 x2=−30872.9954039089 x3=−35111.0096646728 x4=−36806.2158151855 x5=−28330.187841268 x6=−24939.779390143 x7=−26634.9833453947 x8=−35958.6127125654 x9=−19854.1721532085 x10=−14768.5774397478 x11=−17311.3725411122 x12=−13073.3845064641 x13=−24092.177650919 x14=−32568.2008920951 x15=−22396.9747395139 x16=−34263.4066755697 x17=−20701.7727389755 x18=−30025.392786181 x19=−27482.5855320072 x20=0.361524883587661 x21=−19006.57189308 x22=−29177.7902624869 x23=−15616.1751696903 x24=−33415.8037497305 x25=−41891.8353941377 x26=−41044.2320305184 x27=−23244.5760938158 x28=−21549.3736119485 x29=−38501.422170149 x30=−37653.818968837 x31=−16463.7735695936 x32=−12225.7896352151 x33=−18158.9720041999 x34=−40196.6287036936 x35=−39349.0254160413 x36=−13920.9805016283 x37=−31720.5981081316 You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function: Points where there is an indetermination: x1=−1
x→−1−limx+1(x+2)ex−2ex+(x+1)22(xex−1)=∞ x→−1+limx+1(x+2)ex−2ex+(x+1)22(xex−1)=−∞ - the limits are not equal, so x1=−1 - is an inflection point
Сonvexity and concavity intervals: Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points: Concave at the intervals [0.361524883587661,∞) Convex at the intervals (−∞,0.361524883587661]
Vertical asymptotes
Have: x1=−1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo x→−∞lim(x+1xex−1)=0 Let's take the limit so, equation of the horizontal asymptote on the left: y=0 x→∞lim(x+1xex−1)=∞ Let's take the limit so, horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-1 + x*exp(x))/(1 + x), divided by x at x->+oo and x ->-oo x→−∞lim(x(x+1)xex−1)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right x→∞lim(x(x+1)xex−1)=∞ Let's take the limit so, inclined asymptote on the right doesn’t exist
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x). So, check: x+1xex−1=1−x−xe−x−1 - No x+1xex−1=−1−x−xe−x−1 - No so, the function not is neither even, nor odd