Q.Write the answer in one word/sentence: Write the minimum value of $\frac{1-x+x^2}{1+x+x^2}$, for all real values of $x$.

Rational Function Optimization

A rational function is a ratio of two polynomials, $f(x)=\dfrac{p(x)}{q(x)}$ — for example $f(x)=x+\dfrac1x$ or $f(x)=\dfrac{x^2+1}{x}$. Finding its maximum or minimum is a common maxima–minima task, and the only new skill is differentiating a quotient cleanly.

The Method

To optimise $f(x)=\dfrac{p(x)}{q(x)}$:

1. Differentiate with the quotient rule,

2. Set $f'(x)=0$. A fraction is zero only when its numerator is zero, so you only need $p'q-pq'=0$ — the denominator never has to vanish.
3. Respect the domain. Values where $q(x)=0$ are excluded, and many problems restrict to $x>0$. Keep these in mind when choosing which critical point is valid.
4. Classify each critical point with the second-derivative test or a sign check of $f'$.

A Worked Example

Minimise $f(x)=x+\dfrac1x$ for $x>0$.

Differentiating, $f'(x)=1-\dfrac{1}{x^2}$. Setting this to zero gives $x^2=1$, so $x=1$ (taking the positive root, since $x>0$). Then $f''(x)=\dfrac{2}{x^3}$, and $f''(1)=2>0$, confirming a minimum. The minimum value is $f(1)=1+1=2$.

This matches the AM–GM bound $x+\dfrac1x\ge 2$, with equality at $x=1$ — a useful sanity check.

Common Mistakes