Question: "For what values of x is f(x)=2x4+4x3+2x2−2 concave..."

talpajocotefnf3

Answered question

2022-03-27

For what values of x is

f (x) = 2 x^{4} + 4 x^{3} + 2 x^{2} - 2

concave or convex?

Answer & Explanation

yaum3xg1

Beginner2022-03-28Added 12 answers

Step 1
Given function:

f (x) = 2 x^{4} + 4 x^{3} + 2 x^{2} - 2

f^{'} (x) = 8 x^{3} + 12 x^{2} + 4 x

f^{″} (x) = 24 x^{2} + 24 x + 4

The function will be concave when

f^{″} (x) \leq 0

∴ 24 x^{2} + 24 x + 4 \leq

6 x^{2} + 6 x + 1 \leq 0

(x + \frac{3 + \sqrt{3}}{6}) (x + \frac{3 - \sqrt{3}}{6}) \leq 0

x \in (\frac{- 3 - \sqrt{3}}{6}, \frac{- 3 + \sqrt{3}}{6})

Hence, the given function is concave in

(\frac{- 3 - \sqrt{3}}{6}, \frac{- 3 + \sqrt{3}}{6})

the given function is convex in

(- \infty, \frac{- 3 - \sqrt{3}}{6}) \cup (\frac{- 3 + \sqrt{3}}{6}, \infty)

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