How do you determine whether the function $f(x)=(2x-1)^2(x-3)^2$ is concave up or concave down?

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Answer 1 · 2015-09-11T16:09:23

To determine where the graph of $f$ is concave up and where it is concave down, look at the sign of $f''(x)$ .

$f(x)=(2x-1)^2(x-3)^2$

$f'(x) = 2(2x-1)*(2)(x-3)^2 + (2x-1)^2 2(x-3)(1)$

$= 2(2x-1)(x-3)[2(x-3)+(2x-1)]$

$= 2(2x-1)(x-3)(4x-7)$

$f''(x) = 2(2)(x-3)(4x-7)$

$+ 2(2x-1)(1)(4x-7)$

$+2(2x-1)(x-3)(4)$

$= 48x^2-168x+61$

Set $f''(x) = 0$ to find partition numbers: $(21 +- 5sqrt3)/12$

Test each interval:

On $(-oo, (21 - 5sqrt3)/12)$ , $f''$ is positive, graph is concave up

on $(21 - 5sqrt3)/12,(21 + 5sqrt3)/12$ , $f''$ is negative, graph is concave down

on $((21 + 5sqrt3)/12,oo)$ , $f''$ is positive, graph is concave up

How do you determine whether the function f(x)=(2x-1)^2(x-3)^2f(x)=(2x-1)^2(x-3)^2 is concave up or concave down?