talpajocotefnf3

2022-03-27

For what values of x is $f\left(x\right)=2{x}^{4}+4{x}^{3}+2{x}^{2}-2$ concave or convex?

### Answer & Explanation

yaum3xg1

Step 1
Given function:
$f\left(x\right)=2{x}^{4}+4{x}^{3}+2{x}^{2}-2$
${f}^{\prime }\left(x\right)=8{x}^{3}+12{x}^{2}+4x$
$f{}^{″}\left(x\right)=24{x}^{2}+24x+4$
The function will be concave when $f{}^{″}\left(x\right)\le 0$
$\therefore 24{x}^{2}+24x+4\le$
$6{x}^{2}+6x+1\le 0$
$\left(x+\frac{3+\sqrt{3}}{6}\right)\left(x+\frac{3-\sqrt{3}}{6}\right)\le 0$

Hence, the given function is concave in
the given function is convex in

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