Get the solution to "How do you find the intervals where the..."

alahovom7ahc

Answered question

2022-04-10

How do you find the intervals where the function

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

is concave up and concave down?

unduncjineei5r3

Beginner2022-04-11Added 19 answers

Step 1
Concave down in

(- \infty, \frac{13}{6})

and concave up in

(\frac{13}{6}, \infty)

For finding the intervals where f(x) is concave up or concave down, first get f''(x).
f(x) is

(2 x - 1) {(x - 3)}^{2} = 2 x^{3} - 13 x^{2} + 24 x - 9

f^{'} (x) = 6 x^{2} - 26 x + 24

and

f x) = 12 x - 26

. The inflection point is at

x = \frac{13}{6}

Step 2
For

x \leq \frac{13}{6}

, f''(x) is negative, hence f(x) would be concave down in the interval

(- \infty, \frac{13}{6})

For

x \geq \frac{13}{6}

, f''(x) is positive, hence f(x) would be concave up in the interval

(\frac{13}{6}, \infty)

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