How Do You Determine Where a Function is Concave Up or Down?

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In summary, to sketch the graph of y=(4x)/(x^2+1), we first find y'' using the quotient rule, which gives us y'' = (8x^5 - 16x^3 - 24x)/(x^2 + 1)^4. To find the inflection points, we set y'' = 0 and solve for x, which gives us x = 0, x = sqrt(3), and x = -sqrt(3). We ignore the imaginary solution x = sqrt(-1) and use the sign charts to determine that the intervals where the graph is concave up are (-sqrt(3), 0) and (sqrt(3), infinity), and the intervals
  • #1
donjt81
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problem: Use the graphing strategy to sketch the graph of y=(4x)/(x^2+1). check the intervals where it is concave up and where it is concave down. Then graph it. please use sign charts.

to find this we have to first find y''.
so I used the quotient rule twice to get this
y'' = (8x^5 - 16x^3 - 24x)/(x^2 + 1)^4
to find the inflection points we set y'' = 0 and solve for x.
I have a question on this
while solving y'' = (8x^5 - 16x^3 - 24x)/(x^2 + 1)^4 = 0
i come across this step
(8x)(x^2 + 1)(x^2 - 3) = 0
so that would mean

8x = 0 this mean x = 0
(x^2 + 1) = 0 this means x = sqrt(-1)
(x^2 - 3) = 0 this means x = +-sqrt(3)

but do we just ignore the x = sqrt(-1) and conclude that the inflection points are x = 0, x = sqrt(3) and x = -sqrt(3)

So once we get the inflection points we use the sign charts to find concave up and concave down.

intervals where graph is concave up: (-sqrt(3), 0) & (sqrt(3), infinity)
intervals where graph is concave down: (-infinity, -sqrt(3)) & (0, sqrt(3))

are these intervals i found correct?
 
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  • #2
i did not get the imaginary answer you got

i got this for the second derivative
using mathematica
[tex] -\frac{16x}{(x^2+1)^2} +4x\left(\frac{8x^3}{(1+x^2}^3} - \frac{2}{(1+x^2)^3} = 0 [/tex]

it looks like you are assuming the denominator to be zero... taht can't be. that would not make this expression equal to zero.
 

Related to How Do You Determine Where a Function is Concave Up or Down?

1. What is the difference between concave up and concave down?

Concave up and concave down refer to the shape of a curve on a graph. A curve is concave up if it looks like a cup and concave down if it looks like a frown. In other words, a concave up curve is shaped like a smile, while a concave down curve is shaped like a frown.

2. How can you determine if a curve is concave up or concave down?

To determine if a curve is concave up or concave down, you can look at the second derivative of the function. If the second derivative is positive, the curve is concave up. If the second derivative is negative, the curve is concave down.

3. What is the significance of concavity in mathematics?

Concavity is important in mathematics because it helps us understand the behavior of functions. It can tell us whether a function is increasing or decreasing, and can also help us find maximum and minimum points on a curve.

4. Can a curve be both concave up and concave down?

No, a curve cannot be both concave up and concave down at the same time. A curve can only have one type of concavity at any given point. However, a curve can change from being concave up to concave down or vice versa at different points on the graph.

5. How does changing the sign of the second derivative affect the concavity of a curve?

Changing the sign of the second derivative will change the concavity of a curve. If the second derivative is positive, the curve will be concave up. If the second derivative is negative, the curve will be concave down. So, changing the sign of the second derivative will essentially change the shape of the curve on the graph.

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