Inflection of Cosines

Calculus Level 5

How many times does the graph of y = 5 cos ⁡ ( 2 x ) 3 + 3 cos ⁡ ( 2 x ) 5 \large \displaystyle y = \sqrt[3]{5\cos(2x)} + \sqrt[5]{3\cos(2x)} change concavity on the interval from ( − π , π ) (-\pi, \pi) ?

Note : This can be done without taking a second derivative.


The answer is 4.

View wiki
Andrew Ellinor by Andrew Ellinor , 33, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Andrew Ellinor
Oct 15, 2015

So the cool thing about this problem is that you don't need to take a second derivative. Note that the graphs of 5 cos ⁡ ( 2 x ) 5\cos(2x) and 3 cos ⁡ ( 2 x ) 3\cos(2x) have the same periodicity. Moreover, taking any odd root of these functions do not change their periodicity. Lastly, summing them does not change their periodicity. Therefore, we expect as many points of inflection for y = 5 cos ⁡ ( 2 x ) 3 + 3 cos ⁡ ( 2 x ) 5 y = \sqrt[3]{5\cos(2x)} + \sqrt[5]{3\cos(2x)} as we would y = cos ⁡ ( 2 x ) y = \cos(2x) on the interval ( − π , π ) (-\pi, \pi) , which would be 4.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Adding cos ⁡ ( 2 x ) \cos(2x) to cos ⁡ ( x ) \cos(x) does not change its periodicity, therefore cos ⁡ ( x ) + cos ⁡ ( 2 x ) \cos(x)+\cos(2x) would have the same periodicity as cos ⁡ ( x ) \cos(x) , hence also the same number of points of inflection, which is 2.

Kenny Lau - 5 years, 8 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...