I wanted to find a function f(x) that has the following properties: 1.extreme values at x=-1 and x=0 and x=1 2.x -axis is asymptote No other asymptotes or extrema. I found the following two by integrating and trial and error: f1(x)=x2e−x2 f2(x)=2ln⁡(x2+1)+ln⁡(x4−x2+1) How can I find one that is actually a rational function?

Question

I wanted to find a function f(x) that has the following properties:    1.extreme values at x=-1 and x=0 and x=1    2.x -axis is asymptote    No other asymptotes or extrema.    I found the following two by integrating and trial and error:    f1(x)=x2e−x2    f2(x)=2ln⁡(x2+1)+ln⁡(x4−x2+1)    How can I find one that is actually a rational function?

emeriinb4r · Accepted Answer

If the x-axis has to be an asymptote, you need a denominator of higher degree than the numerator. In order to get the extreme values where you want them, suggest a rational function with sufficient parameters, e.g. suggest a function of the form: f(x)=ax2+bx+cx4+1 Note that taking x4+1 instead of x3+1(which is of sufficiently high degree as well) ensures no vertical asymptote is added. Finding f′(x) and requiring f′(-1)=f′(0)=f′(1)=0 yields b=c=0 suggesting that any function with non-zero parameter a of the following form will do: d(x)=ax2x4+1 You can verify that indeed, taking e.g. a=1, this function satisfies your requirements.

Nathan Kent · Accepted Answer

You must have:    f′(-1)=f′(1)=f′(0)=0    So the numerator of the derivative of your function must have those 3 zeros.    limx→∞f(x)=0,    suggesting the denominator of the rational function must have higher degree than the numerator.

I wanted to find a function f(x) that has the

Answered question

Answer & Explanation

New Questions in Algebra II