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

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In summary, the conversation is about finding the concavity of f(x)=(2x)/((x^2)-25) and the attempt at a solution involved finding the second derivative, but the results were incorrect. The correct concavity of the function is that it is concave down for all values smaller than 0 and concave up for all values larger than 0, with the exception of x=-5 and x=5.
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Homework Statement



f(x)=(2x)/((x^2)-25)

find concave up and down

Homework Equations


The Attempt at a Solution



I found the second derivative to b

-4x((-2x^2)-24)
-----------------
((x^2)-25)^2

i found the only inflection point was x=0 (which was correct)

I plugged in values on both the right and left side of 0 and determined that f(x) was concave down on all values smaller than 0 with the exception of -5, and f(x) was concave up on all values larger than 0 with the exception of 5

however this is not the correct answer. Can someone confirm/tell me what i am doing wrong?

thanks
 
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Something is wrong with your calculations as that is not correct for the second derivative.
 

FAQ: How to Determine Where a Function is Concave Up or Down?

What is the definition of concave up and down?

Concave up and down are terms used to describe the shape of a curve or function. A function is concave up if it is shaped like a cup, with the bottom of the cup being the lowest point. Conversely, a function is concave down if it is shaped like a hill, with the peak of the hill being the highest point.

How do you find the concavity of a function?

To find the concavity of a function, you can use the second derivative test. Take the second derivative of the function and evaluate it at a specific point. If the second derivative is positive, the function is concave up at that point. If the second derivative is negative, the function is concave down at that point.

What does concave up and down tell us about a function?

The concavity of a function provides information about the rate of change of the function. A concave up function has a positive rate of change, meaning it is increasing. A concave down function has a negative rate of change, meaning it is decreasing.

Can a function be both concave up and down?

No, a function can only be concave up or concave down at a specific point. It cannot be both at the same time. However, a function can change concavity at different points on its graph.

How do you graph a function with concave up and down?

To graph a function with concave up and down, you can start by finding the concavity using the second derivative test. Then, plot the points where the concavity changes and connect them with a smooth curve. Keep in mind that the function will be increasing when concave up and decreasing when concave down.

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