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?

yaum3xg1

Beginner2022-03-28Added 12 answers

Step 1

Given function:

$f\left(x\right)=2{x}^{4}+4{x}^{3}+2{x}^{2}-2$

${f}^{\prime}\left(x\right)=8{x}^{3}+12{x}^{2}+4x$

$f{}^{\u2033}\left(x\right)=24{x}^{2}+24x+4$

The function will be concave when$f{}^{\u2033}\left(x\right)\le 0$

$\therefore 24{x}^{2}+24x+4\le$

$6{x}^{2}+6x+1\le 0$

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

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

Hence, the given function is concave in$(\frac{-3-\sqrt{3}}{6},\text{}\frac{-3+\sqrt{3}}{6})$

the given function is convex in$(-\mathrm{\infty},\text{}\frac{-3-\sqrt{3}}{6})\cup (\frac{-3+\sqrt{3}}{6},\text{}\mathrm{\infty})$

Given function:

The function will be concave when

Hence, the given function is concave in

the given function is convex in

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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