For what values of x is $f (x) = (x - 2) (x - 4) (x - 3)$ $f(x)=(x-2)(x-4)(x-3)$ concave or convex?

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Answer 1 · 2016-11-22T03:21:31

$f (x)$ $f(x)$ is concave on (-oo,3), and convex on $(3, \infty)$ $(3,oo)$ .

To find where a function is concave or convex, find where $f''(x)$ is positive or negative (respectively).

$f(x)=(x-2)(x-4)(x-3)$

$f'(x)=(1)(x-4)(x-3)+(x-2)(1)(x-3)+(x-2)(x-4)(1)$
$f'(x)=x^2-7x-12+x^2-5x+6+x^2-6x+8$
$f'(x)=3x^2-18x+2$

$f''(x)=6x-18$
$f''(x)=6(x-3)$

$f''(x)$ is negative from $(-oo,3)$ , and positive from $(3,oo)$

Therefore, $f(x)$ is concave on (-oo,3), and convex on $(3,oo)$ .

For what values of x is f(x)=(x-2)(x-4)(x-3)f(x)=(x−2)(x−4)(x−3)f(x)=(x-2)(x-4)(x-3) concave or convex?