What are the local extrema, if any, of $f (x) = x^{3} - 12 x + 2$ $f (x) = x^3-12x+2$ ?

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Answer 1 · 2015-11-09T16:55:16

The function has 2 extrema:

$f_{max}(-2)=18$ and $f_{min}(2)=-14$

We have a function: $f(x)=x^3-12x+2$

To find extrema we calculate derivative

$f'(x)=3x^2-12$

The first condition to find extreme points is that such points exist only where $f'(x)=0$

$3x^2-12=0$

$3(x^2-4)=0)$

$3(x-2)(x+2)=0$

$x=2 vv x=-2$

Now we have to check if the derivative changes sign at the calcolated points:

graph{x^2-4 [-10, 10, -4.96, 13.06]}

From the graph we can see that $f(x)$ has maximum for $x=-2$ and minimum for $x=2$ .

Final step is to calculate the values $f(-2)$ and $f(2)$

What are the local extrema, if any, of f (x) = x^3-12x+2 f(x)=x3−12x+2f (x) = x^3-12x+2 ?