Exponential Functions

[Onekeeper]

Hi, Is anyone able to give me a hand solving this question?

Find the stationary point on the curve y=xe^x and determine its nature. Find any points of inflexion and find values of y as x becomes very large or small. Hence sketch the curve.


Thanks in advance!

[RuiAce]

\begin{align*}y&=xe^x\\ \frac{dy}{dx} &= (1+x)e^x\\ \frac{d^2y}{dx^2} &= (2+x)e^x\end{align*}

Then just plug in some large values such as \(x = 1000\) onto the calculator and see what you get. (You should find something close to 0 when you plug something like \(x=-1000\), which indicates an horizontal asymptote as you go towards the left.) Much of this is very standard and you're expected to know how to do it so if you need further details clarified you should make it clear where you are having trouble.