Change in concavity of a function

[European Sens]

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

 

[sutupidmath]

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.

 

[European Sens]

I think its E.

 

[sutupidmath]

I think its E.

Why is that so?

 

[sutupidmath]

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.

 

[Feldoh]

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]