Question

What are the maximum and minimum values that the function `f(x)=x/(1 + x^2)`?

Answer 1

Maximum: `1/2`
Minimum: `-1/2`

Explanation:

An alternative approach is to rearrange the function into a quadratic equation. Like this:

`f(x)=x/(1+x^2)rarrf(x)x^2+f(x)=xrarrf(x)x^2-x+f(x)=0`

Let `f(x)=c" "` to make it look neater :-)

`=> cx^2-x+c=0`

Recall that for all real roots of this equation the discriminant is positive or zero

So we have, `(-1)^2-4(c)(c)>=0" "=>4c^2-1<=0" "=>(2c-1)(2c+1)<=0`

It is easy to recognise that `-1/2<=c<=1/2`

Hence, `-1/2<=f(x)<=1/2`

This shows that the maximum is `f(x)= 1/2` and the minimum is `f(x)=1/2`