Concavity of y = x(cosx) at x = pi/3: Second Derivative Test

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In summary, the second derivative test is a method used to determine the concavity of a function at a specific point by analyzing the sign of the second derivative of the function at that point. The second derivative of a function can be found by taking the derivative of the first derivative of the function. Using the second derivative test, we can find that the second derivative of y = x(cosx) at x = pi/3 is equal to -1. The concavity of a function affects the shape of its graph and can also affect the direction of the function's slope at a specific point. Other methods for determining concavity include the first derivative test and the use of the graph of the function, but the second derivative test is often preferred for its
  • #1
donjt81
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I am doing this problem and I am getting stuck at solving the equation

problem: Use the second derivative test to determine the concavity of the following function. y = x(cosx) at x = pi/3

solution: y' = -xsinx + cosx
y'' = -xcosx - 2sinx = 0

and then i did
-xcosx = 2sinx ( i don't know if this is correct)
and then I am stuck... i don't know how to proceed.

please help
 
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  • #2
The function is upward concave is f''(x) > 0, and downward concave if f''(x) < 0. So just plug x=pi/3 into your function for y''.
 
  • #3
ohhhh! duhhh! I shouldve known... thanks
 

1. What is the second derivative test and how is it used to determine the concavity of a function at a specific point?

The second derivative test is a method used to determine the concavity of a function at a specific point by analyzing the sign of the second derivative of the function at that point. A positive second derivative indicates a concave up shape, while a negative second derivative indicates a concave down shape.

2. How do you find the second derivative of a function?

The second derivative of a function can be found by taking the derivative of the first derivative of the function. In other words, it is the derivative of the slope of the original function.

3. What is the concavity of y = x(cosx) at x = pi/3?

Using the second derivative test, we can find that the second derivative of y = x(cosx) at x = pi/3 is equal to -1. This indicates a concave down shape at x = pi/3.

4. How does the concavity of a function affect its graph?

The concavity of a function affects the shape of its graph. A concave up shape will have a "smiling" appearance, while a concave down shape will have a "frowning" appearance. This can also affect the direction of the function's slope at a specific point.

5. Are there any other methods for determining the concavity of a function?

Yes, there are other methods such as the first derivative test and the use of the graph of the function. However, the second derivative test is often preferred as it provides a more accurate and definitive result.

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