How many critical points does the function 𝑔(𝑥)=|1−𝑥2|𝑔(𝑥)=|1−𝑥2| have?

Question

How many critical points does the function g(x) =  |1 - x²| have?

Answer 1

Option #1: replace |z| = sqrt(z^2)

g(x) = | 1 - x^2 | = sqrt ( (1-x^2)^2)

= ((1-x^2)^2)^(1/2)

0= g'(x) = (1/2) ((1 - x^2)^2)^(-1/2) (2(1-x^2)(-2x))

= (-2x)(1 - x^2) / ((1 - x^2)^2)^1/2

= (-2x)(1-x^2)/ (1 - x^2)

= -2x

relative max at (0,1)

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Option #2: solve the problem twice...

1st, as is; then again with x=-x

it turns out that it is an EVEN function that eats

the negative....

g(-x) = | 1 - (-x)^2 | = | 1 - x^2 | = g(x)

so the absolute values can be dropped because the

function is perfectly symmetrical..

the derivative is -2x , as shown above..

relative max is at ( 0,1)