For what values of x is $f (x) = (x - 1) (x - 6) (x - 2)$ $f(x)=(x-1)(x-6)(x-2)$ concave or convex?

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Answer 1 · 2017-02-20T14:08:03

See below.

$f (x) = (x - 1) (x 6) (x - 2) = x^{3} - 9 x^{2} + 20 x - 12$ $f(x) = (x-1)(x6)(x-2) = x^3-9x^2+20x-12$

$f'(x) = 3x^2-18x+20$

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

$f''(x)$ is negative for $x < 3$ , so $f$ is concave (or concave down) on $(-oo,3)$ .

$f''(x)$ is positive for $3 < x$ , so $f$ is convex (or concave up) on $(3,oo)$ .

For what values of x is f(x)=(x-1)(x-6)(x-2)f(x)=(x−1)(x−6)(x−2)f(x)=(x-1)(x-6)(x-2) concave or convex?