I need help to plot the function $x^2+(1-x)^8\sin\left(\frac{1}{1-x}\right)$ in the range $[0.5,1)$ together with the functions $x^2+(1-x)^8$ and $x^2-(1-x)^8$.
As $x$ approaches $1$ the first function is supposed to oscillate between the two other functions, but I am not able to capture this effect in a plot. The reason for this is that the absolute value of $(1-x)^8$ becomes very small. Are you able to plot these three functions and observe the oscillation?
For those interested in the motivation of this function: I wanted a function that does not behave nicely, but that is positive, has a positive first derivative, and has a positive second derivative on the range $[0.5, 1]$(here we require that the funciton value is $1$ at $x=1$.)