Find concave up and concove down of function f(x) = x^2 ∣x−3∣


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Question Number 121956 by bemath last updated on 13/Nov/20

$F i n d c o n c a v e u p a n d c o n c o v e d o w n$ $o f f u n c t i o n f (x) = x^{2} ∣ x - 3 ∣$
	Answered by MJS_new last updated on 13/Nov/20
	$f(x)= { ((3x^2 −x^3 ; x<3)),((x^3 −3x^2 ; x≥3)) :} f′′(x)= { ((6−6x; x<3)),((6x−6; x≥3)) :} ⇒ f(x) is concave { ((up; x<1)),((down; 1<x<3)),((up; 3<x)) :}$
	$f (x) = {\begin{cases} 3 x^{2} - x^{3}; x < 3 \\ x^{3} - 3 x^{2}; x ⩾ 3 \end{cases}$ $f^{″} (x) = {\begin{cases} 6 - 6 x; x < 3 \\ 6 x - 6; x ⩾ 3 \end{cases}$ $\Rightarrow$ $f (x) is concave {\begin{cases} up; x < 1 \\ down; 1 < x < 3 \\ up; 3 < x \end{cases}$
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