2. Determine the intervals on which f (x)= is concave up and concave down. .x* +1Thenstate the inflection points.state so. Round answers to 2 decimal places, if needed.If there or no intervals of concavity or no inflection points, then

Answered: 2. Determine the intervals on which f…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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Related questions

Question

2. Determine the intervals on which f (x)= is concave up and concave down. .
x* +1
Then
state the inflection points.
state so. Round answers to 2 decimal places, if needed.
If there or no intervals of concavity or no inflection points, then

Expert Solution

Step 1 Given

$f (x) = \frac{x}{x^{2} + 1}$

First find the inflection point of given function

An inflection point is a point on the graph at which the second derivative changes sign.

If $f'' (x) > 0$ then $f (x)$ concave upwards.

If $f'' (x) < 0$ then $f (x)$ concave downwards.

Find derivative of $f (x)$ with respect to $x$ .

$\frac{d}{d x} (f (x)) = \frac{d}{d x} (\frac{x}{x^{2} + 1})$

Apply the quotient rule

${(\frac{f}{g})}^{'} = \frac{f' g - g' f}{g^{2}}$

$f (x) = x, g (x) = x^{2} + 1$

$f' (x) = 1, g' (x) = 2 x$

Plug all this values in given formula

$\begin{array}{rcl} \frac{d}{d x} (\frac{x}{x^{2} + 1}) & = & \frac{1 (x^{2} + 1) - 2 x (x)}{{(x^{2} + 1)}^{2}} \\ = & \frac{x^{2} + 1 - 2 x^{2}}{{(x^{2} + 1)}^{2}} \\ f' (x) & = & \frac{1 - x^{2}}{{(x^{2} + 1)}^{2}} \end{array}$

Find second derivative

$f'' (x) = \frac{- 2 x (- x^{2} + 3)}{{(x^{2} + 1)}^{3}}$

Step by step

Solved in 2 steps

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
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2. Determine the intervals on which f (x)= is concave up and concave down. . x* +1 Then state the inflection points. state so. Round answers to 2 decimal places, if needed. If there or no intervals of concavity or no inflection points, then