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 Solution is not found, it's possible that the graph doesn't intersect 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). 10e0+1 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 The roots of this equation x1=0 The values of the extrema at the points:
(0, 1)
Intervals of increase and decrease of the function: Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from: Minima of the function at points: x1=0 The function has no maxima Decreasing at intervals [0,∞) Increasing at intervals (−∞,0]
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=−33415.8037497305 x2=−28330.187841268 x3=−38501.422170149 x4=−23244.5760938158 x5=−30872.9954039089 x6=−19854.1721532085 x7=−24092.177650919 x8=−18158.9720042003 x9=−37653.818968837 x10=−30025.392786181 x11=−26634.9833453947 x12=−17311.3725411143 x13=−29177.7902624869 x14=−16463.7735696055 x15=−35958.6127125654 x16=−34263.4066755697 x17=−22396.9747395139 x18=−19006.5718930801 x19=−32568.2008920951 x20=−39349.0254160413 x21=−35111.0096646728 x22=−20701.7727389755 x23=−13073.3845178082 x24=−12225.7896953671 x25=−41044.2320305184 x26=−31720.5981081316 x27=−25787.3812935264 x28=−21549.3736119485 x29=−15616.1751697581 x30=−14768.577440128 x31=−40196.6287036936 x32=−13920.9805037239 x33=−41891.8353941377 x34=−24939.779390143 x35=−36806.2158151855 x36=−27482.5855320072 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: Have no bends at the whole real axis
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