What are the critical points of $f (x) = x^{3} - 12 x + 7$ $f(x) = x^3 − 12x + 7$ ?

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Answer 1 · 2016-01-07T12:09:56

The critical points are:

MIN: $(2; - 9)$ $(2;-9)$
MAX: $(- 2; 23)$ $(-2;23)$

The critical points are the x values where:

$f'(x)=0$

We obtain:

$f'(x)=3x^(3-1)-12x^(1-1)+0=3x^2-12=3*(x^2-4)=$
$=3*(x+2)(x-2)$

Then:

$f'(x)=0 iff 3*(x+2)(x-2)=0$

a multipication is zero if the factors are zero:

$x+2=0$

$x_1=-2$

$x-2=0$

$x_2=+2$

Now, to find if they are min or max you have to evaluate $f'(x)>0$

$f'(x)>0$ for $x in ]-oo;-2[ uu ]2;+oo[$

then

$f(x)$ is growing for $x in ]-oo;-2[ uu ]2;+oo[$

$f'(x)<0$ for $x in ]-2;2[$

then

$f(x)$ is decreasing for $x in ]-2;2[$

then
for $x=-2$ we have a MAX
for $x=2$ we have a MIN

$f(-2)=-8+24+7=23$

$f(2)=8-24+7=-9$

What are the critical points of f(x) = x^3 − 12x + 7f(x)=x3−12x+7f(x) = x^3 − 12x + 7?