How do I find all the critical points of $f (x) = {(x - 1)}^{2}$ $f(x)=(x-1)^2$ ?

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How do I find all the critical points of $f (x) = {(x - 1)}^{2}$ $f(x)=(x-1)^2$ ?

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Answer 1 · 2015-10-29T15:42:23

See the explanation.

Explanation:

The critical numbers for a function $f$ $f$ and the numbers in the domain of $f$ $f$ , at which either $f (x)$ $f(x)$ does not exist or $f (x) = 0$ $f(x)=0$ .

For $f (x) = {(x - 1)}^{2}$ $f(x)=(x-1)^2$ , note that the domain is $(- \infty, \infty)$ $(-oo,oo)$ .

Find $f'(x)$
We can use the power and chain rule to get:

$f'(x) = 2(x-1)^1 d/dx(x-1) = 2(x-1)(1) = 2(x-1)$

Or we can expand $f(x) = x^2-2x+1$ and differentiate this polynomial:

$f'(x) = 2x-2$
Find the critical numbers for $f$

Either way, we see that
$f'(x) is never undefined and$ f'(x) = 0 $at$ x=1#.

$1$ is in the domain of $f$ , so it is a critical numbers for $f$ .

The "critical points" are (most commonly), the same as critical numbers.
But some use a variant terminology in which critical points are points on the graph, so the have both $x$ and $y$ coordinates.

If you are in such a situation, you need to find $y$ when $x=1$ to finish.

$y=f(1) = 0$ , so the point on the graph is $(1,0)$

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How do I find all the critical points of $f (x) = {(x - 1)}^{2}$ $f(x)=(x-1)^2$ ?

1 Answer

Explanation:

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How do I find all the critical points of f(x)=(x-1)^2f(x)=(x−1)2f(x)=(x-1)^2?

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Explanation:

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How do I find all the critical points of $f (x) = {(x - 1)}^{2}$ $f(x)=(x-1)^2$ ?