# Thread: Concave up and Concave down

1. ## Concave up and Concave down

Hey

I don't understand how to determine the domain of concave up versus concave down when the graph has two points of inflection.

For example

x^4 + 8x^3 + 18x^2 + 8

extreme points: -3, 0
(graph is high at -3 and low at 0)

Points of inflection: -3, -1

My textbook says the answer is
concave up (- infinity , -3) , ( -1, inifinity)
Concave down ( -3 , -1)

2. Originally Posted by Mudd_101
Hey

I don't understand how to determine the domain of concave up versus concave down when the graph has two points of inflection.

For example

x^4 + 8x^3 + 18x^2 + 8

extreme points: -3, 0
(graph is high at -3 and low at 0)

Points of inflection: -3, -1

My textbook says the answer is
concave up (- infinity , -3) , ( -1, inifinity)
Concave down ( -3 , -1)
$\displaystyle f(x) = x^4 + 8x^3 + 18x^2 + 8$

$\displaystyle f'(x) = 4x^3 + 24x^2 + 36x$

$\displaystyle f'(x) = 4(x^3 + 6x^2 + 9x)$

$\displaystyle f''(x) = 4(3x^2 + 12x + 9) = 12(x^2 + 4x + 3)$

set $\displaystyle f''(x) = 0$ ...

$\displaystyle 12(x + 3)(x + 1) = 0$

$\displaystyle x = -3$, $\displaystyle x = -1$

for $\displaystyle x < -3$, $\displaystyle f''(x) > 0$ ... $\displaystyle f(x)$ is concave up

for $\displaystyle -3 < x < -1$, $\displaystyle f''(x) < 0$ ... $\displaystyle f(x)$ is concave down

for $\displaystyle x > -1$ , $\displaystyle f''(x)> 0$ ... $\displaystyle f(x)$ is concave up