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

Question

maggionmoo · Accepted Answer

Answer: To determine where the graph of f is concave up and where it is concave down, look at the sign of f''(x).
Step 1

f (x) = {(2 x - 1)}^{2} {(x - 3)}^{2}

f^{'} (x) = 2 (2 x - 1) \cdot (2) {(x - 3)}^{2} + {(2 x - 1)}^{2} 2 (x - 3) (1)

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

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

f^{″} (x) = 2 (2) (x - 3) (4 x - 7)

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

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

Step 2
Set

f x) = 0

to find partition numbers:

\frac{21 \pm 5 \sqrt{3}}{12}

Test each interval:
On

(- \infty, \frac{21 - 5 \sqrt{3}}{12})

, f" is positive, graph is concave up
on

\frac{21 - 5 \sqrt{3}}{12}, \frac{21 + 5 \sqrt{3}}{12}

, f" is negative, graph is concave down
on

(\frac{21 + 5 \sqrt{3}}{12}, \infty)

, f" is positive, graph is concave up

How do you determine whether the function