Why can you use the square of an area when differentiating to determine the maximum and minimum value

Question

my lecturer recently showed us that when finding a maximum or minimum value of a rectangle, if the equation is too tedious to differentiate normally, we can instead take the square of the equation instead to differentiate.

The Equation was this,

Let r be a constant, $$ A=4x\sqrt{r^2-x^2} $$ He then said this would be too time consuming to differentiate normally in an exam and squared the whole function before differentiating. $$ A^2=16x^2(r^2-x^2) $$ $$ W=A^2 $$ $$ f'(W)=32r^2-64x^3 $$

He did not explain why you could do this but I am curious about why in this case it is alright to do this?

Answer 1

This works because for $A$ greater than or equal to $0$, $A^2$ is a monotonically increasing function of $A$. If you find parameters that maximize/minimize $A^2$, they will also maximize/minimize $A$.

You could in theory apply any monotonic transformation like multiplication, a logarithm, or exponentiation, the square just happens to make the derivative more convenient.

Answer 2

First reason if the square of a function has an extremum, the functin itself has also. Formal reason:: if you differentiate,the squaroot goes to the nominator, The derivative of the Square is in the denominator which has to be zero.