Reuben Brennan

2022-03-26

Determine whether the function $f\left(x\right)=\frac{x}{{x}^{2}+2}$ is concave up or concave down and its intervals?

clarkchica44klt

Beginner2022-03-27Added 17 answers

Step 1

We have that$f\left(x\right)=\frac{x}{{x}^{2}+2}$ calculating its second derivative we find that

$d}^{2}\frac{f\left(x\right)}{{d}^{2}x}=\frac{2x({x}^{2}-6)}{{({x}^{2}+2)}^{3}$

So we need to see how the signs change of$2x({x}^{2}-6)$ as x goes from $-\mathrm{\infty}$ to $+\mathrm{\infty}$

So from$(-\mathrm{\infty},\text{}-\sqrt{6}]$ we have that $f{}^{\u2033}\left(x\right)<0$

from$[-\sqrt{6},\text{}0]$ we have that $f{}^{\u2033}\left(x\right)>0$

from$[0,\text{}\sqrt{6}]$ we have that $f{}^{\u2033}\left(x\right)<0$

from$[\sqrt{6},\text{}+\mathrm{\infty})$ we have that $f{}^{\u2033}\left(x\right)>0$

In order to determine concavity we use the following theorem

Concavity Theorem:

If the function f is twice differentiable at$x=c$ , then the graph of f is concave upward at $(c;\text{}f\left(c\right))$ if $f{}^{\u2033}\left(c\right)>0$ and concave downward if $f{}^{\u2033}\left(c\right)<0$

We have that

So we need to see how the signs change of

So from

from

from

from

In order to determine concavity we use the following theorem

Concavity Theorem:

If the function f is twice differentiable at

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

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?