Change in concavity of a function

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Discussion Overview

The discussion centers on determining how many times the function f(x) = x^2 + 5cosx changes concavity within the interval 0 <= x <= 2*pi*n, where n is a positive integer. The focus includes mathematical reasoning and exploration of inflection points.

Discussion Character

  • Mathematical reasoning, Exploratory

Main Points Raised

  • Some participants propose that the function changes concavity E) 2n times.
  • Others suggest that the number of changes in concavity can be determined by finding points where the function is concave up or down.
  • A participant notes that changes in concavity occur at inflection points, leading to the consideration of the second derivative f''(x) = 2 - 5cosx.
  • It is mentioned that the equation 0 = 2 - 5cosx can be solved to find when cosx = 2/5, which is relevant for determining the inflection points.
  • Participants are encouraged to analyze how many times this condition is satisfied over the interval [0, 2pi*n].

Areas of Agreement / Disagreement

There is no consensus on the number of times the function changes concavity, as participants have differing views on the answer and the method of determining it.

Contextual Notes

The discussion involves assumptions about the behavior of the cosine function and its periodicity, as well as the implications of the second derivative test for concavity. The exact number of inflection points is not resolved.

Who May Find This Useful

Readers interested in calculus, particularly in the study of concavity and inflection points, may find this discussion relevant.

European Sens
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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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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.
 
I think its E.
 
European Sens said:
I think its E.
Why is that so?
 
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.
 
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]
 

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