Change in concavity of a function

In summary, the function f(x) = x^2 + 5cosx changes concavity 2n times in the interval 0 <= x <= 2*pi*n, where n is a positive integer. This is determined by finding the points at which the second derivative of the function is equal to 0, which occur at cosx = 2/5.
  • #1
European Sens
19
0
If n is a positive integer, how many times does the function f(x) = x^2 + 5cosx change concavity in the interval 0 <= x <= 2*pi*n?

A) 0
B) 1
C) 2
D) n
E) 2n
 
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  • #2
European Sens said:
If n is a positive integer, how many times does the function f(x) = x^2 + 5cosx change concavity in the interval 0 <= x <= 2*pi*n?

A) 0
B) 1
C) 2
D) n
E) 2n

What do you think?

And welcome to PF.
 
  • #3
I think its E.
 
  • #4
European Sens said:
I think its E.
Why is that so?
 
  • #5
hint: find all points at which that function is concave up and concave down and see if you can determine how many times it changes it's concavity.
 
  • #6
The function changes concavity at an inflection point so we take the second derivative.

f''(x) = 2-5cosx

0 = 2-5cosx

So whenever cosx = 2/5 there is a change in inflection. Then just figure out how many times that happens on [0,2pi*n]
 

What is a change in concavity of a function?

A change in concavity of a function refers to a change in the direction of the curvature of the graph of the function. It can be either a change from concave up to concave down, or vice versa.

How is the concavity of a function determined?

The concavity of a function is determined by the second derivative of the function. If the second derivative is positive, the function is concave up, and if the second derivative is negative, the function is concave down.

What does a point of inflection represent in terms of concavity?

A point of inflection is a point on the graph of a function where the concavity changes. It marks the transition from concave up to concave down, or vice versa.

What is the significance of a change in concavity of a function?

A change in concavity can indicate a change in the behavior of the function. It can also help to identify key points on the graph, such as local maxima or minima.

How can a change in concavity affect the shape of a function's graph?

A change in concavity can result in a change in the shape of a function's graph. For example, a change from concave up to concave down can result in the function having a point of inflection or a local maximum or minimum.

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