The vertices of a regular polygon of 12 sides are joined to form triangles. Find the number of triangles which do not have their sides as the sides of the polygon

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Solution

$x^{3} - 3 x + k = 0 f (x) = x^{3} - 3 x$
$g (x) = - k$
$f (x) = x^{3} - 3 x$
$f^{'} (x) = 3 x^{2} - 3 = 0 x = \pm 1$
$f (- 1) = 3 - 1 = 2$
$f (1) = 1 - 3 = - 2$
$k > 2 or k < - 2$
$f (x)$ has exactly one root.
$| k | > 2 Min. value of [| k |] = 2$

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The vertices of a regular polygon of 12 sides are joined to form triangles. Find the number of triangles which do not have their sides as the sides of the polygon

x3–3x+k=0 f(x)=x3–3x g(x)=–k f(x)=x3–3x f′(x)=3x2–3=0 x=±1 f(–1)=3–1=2 f(1)=1–3=–2 k>2 or k<–2 f(x) has exactly one root. |k|>2 Min. value of [|k|]=2

$x^{3} - 3 x + k = 0 f (x) = x^{3} - 3 x$
$g (x) = - k$
$f (x) = x^{3} - 3 x$
$f^{'} (x) = 3 x^{2} - 3 = 0 x = \pm 1$
$f (- 1) = 3 - 1 = 2$
$f (1) = 1 - 3 = - 2$
$k > 2 or k < - 2$
$f (x)$ has exactly one root.
$| k | > 2 Min. value of [| k |] = 2$