Find the absolute minimum and absolute maximum

Question

Find the absolute minimum and absolute maximum in [-1,2] of the function $ \ f(x)=|x| \ $. $$ $$ I have got absolute minimum at x=0 , ie., $ f(0)=0 $ is the absolute mimimum. The absolute maximum is $ f(2)=2 $ . Is this correct ?

Answer 1

We can't differentiate normally when there is a modulus sign, so consider $g(x)=f(x)^2=|x|^2=x^2$

Then $g'(x) =2x$

Setting $g'(x)=0\implies f(x)^2=0\implies f(x)=0$ so clearly a minimum there.

Then we can maximise $g(x)$ by testing the upper and lower bounds of the given domain i.e. $g(-1)$ and $g(2)$.

$g(-1)=1<g(2) =4$. Thus $g(x)$ is maximised as $x=2$, which implies that $f(x)$ is maximised at $x=2$, so $f_{\text{max}}=f(2)=2$

Alternatively you could consider the separate cases when $f(x)$ is positive or negative and compute the derivatives accordingly, however the method I have used is often more helpful when dealing with more difficult questions.