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+1x+ex=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 (x + exp(x))/(1 + x). 1e0 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+1ex+1−(x+1)2x+ex=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+1ex−x+12(ex+1)+(x+1)22(x+ex)=0 Solve this equation The roots of this equation x1=−33415.8037497305 x2=−28330.187841268 x3=−38501.422170149 x4=−23244.5760938158 x5=−15616.1751697251 x6=−30872.9954039089 x7=−24092.177650919 x8=0.483259389716171 x9=−37653.818968837 x10=−19854.1721532085 x11=−30025.392786181 x12=−26634.9833453947 x13=−19006.5718930801 x14=−29177.7902624869 x15=−35958.6127125654 x16=−18158.9720042001 x17=−34263.4066755697 x18=−20701.7727389755 x19=−22396.9747395139 x20=−13920.98050271 x21=−32568.2008920951 x22=−39349.0254160413 x23=−35111.0096646728 x24=−41044.2320305184 x25=−31720.5981081316 x26=−25787.3812935264 x27=−21549.3736119485 x28=−17311.3725411133 x29=−40196.6287036936 x30=−41891.8353941377 x31=−24939.779390143 x32=−13073.3845123319 x33=−36806.2158151855 x34=−16463.7735695997 x35=−14768.5774399437 x36=−27482.5855320072 x37=−12225.7896664061 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+1ex−x+12(ex+1)+(x+1)22(x+ex)=∞ x→−1+limx+1ex−x+12(ex+1)+(x+1)22(x+ex)=−∞ - 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 [0.483259389716171,∞) Convex at the intervals (−∞,0.483259389716171]
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+1x+ex)=1 Let's take the limit so, equation of the horizontal asymptote on the left: y=1 x→∞lim(x+1x+ex)=∞ 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 (x + exp(x))/(1 + x), divided by x at x->+oo and x ->-oo x→−∞lim(x(x+1)x+ex)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right x→∞lim(x(x+1)x+ex)=∞ 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+1x+ex=1−x−x+e−x - No x+1x+ex=−1−x−x+e−x - No so, the function not is neither even, nor odd