# For what values of x is f(x)=(7x-1)(x-6)(x-2) concave or convex?

Nov 4, 2016

$f \left(x\right) = \left(7 x - 1\right) \left(x - 6\right) \left(x - 2\right)$ is concave down on the interval $x < \frac{19}{7}$, and $f \left(x\right)$ is concave up on the interval $x > \frac{19}{7}$.

#### Explanation:

$f \left(x\right) = \left(7 x - 1\right) \left(x - 6\right) \left(x - 2\right)$
Find first derivative:
$f ' \left(x\right) = 21 {x}^{2} - 114 x + 92$
Find second derivative:
$f ' ' \left(x\right) = 42 x - 114$
$f ' ' \left(x\right) = 6 \left(7 x - 19\right)$
Find where second derivative is negative to get intervals of concave down, and where $f ' ' \left(x\right)$ is positive is where $f \left(x\right)$ is concave up.
$f ' ' \left(x\right)$ is positive when $x > \frac{19}{7}$, and $f ' ' \left(x\right)$ is negative when $x < \frac{19}{7}$.