# A Very Unique Maxima - Minima problem in Differential calculus

• July 12th 2011, 09:52 AM
amahindroo
A Very Unique Maxima - Minima problem in Differential calculus
Hi All,

I have a very unique maxima-minima problem in differential calculus:-

Find the point of absolute maxima and minima for the function given as :-

y = x + sin(2x), 0 <= x <= 2(pi)

Also find the respective maximum and minimum values.

It seems to be a very simple problem which can be easily solved by the Second Derivative test. the steps that I have already carried out are:-

y = x + sin(2x) = x + 2sin(x)cos(x)
dy/dx = 1 + 2 (cos^2(x) - sin^2(x)) = 1 + 2(2cos^2(x) - 1) = 4cos^2(x) - 1

For y to be maximum or minimum, dy/dx = 0
4cos^2(x) - 1 = 0
4cos^2(x) = 1
cos^2(x) = (1/4)
cos(x) = + (1/2) or - (1/2)
x = (pi)/3, (5(pi))/3 or x = (2(pi))/3, (4(pi))/3

d2y/dx2 = -8cos(x)sin(x)

So, d2y/dx2 at x=(pi)/3 is = -8X(+ve)X(+ve) < 0
So, d2y/dx2 at x=5(pi)/3 is = -8X(+ve)X(-ve) > 0
So, d2y/dx2 at x=2(pi)/3 is = -8X(-ve)X(+ve) > 0
So, d2y/dx2 at x=4(pi)/3 is = -8X(-ve)X(-ve) < 0

So, according to this, x = (pi)/3 and 4(pi)/3 needs to be the points of local maxima and one of them to be the absolute maxima

Similarly,
x = 2(pi)/3 and 5(pi)/3 needs to be the points of local minima and one of them to be the absolute minima

But the answer key for the exercise containing the above question shows x = 2(pi) to be the point of maxima and 2(pi) to be the maximum value and x = 0 to be the point of minima with 0 as the minimum value

This answer (x = 2(pi) and x = 0) if we put the values in the function y = x + sin(2x) seems to be the logical solution but we are not able to arrive at it using derivatives.

Can anyone help me find the solution to this problem? or is this a limitation of finding a solution using derivatives?

amahindroo
• July 12th 2011, 10:11 AM
Plato
Re: A Very Unique Maxima - Minima problem in Differential calculus
Quote:

Originally Posted by amahindroo
Find the point of absolute maxima and minima for the function given as
$y = x + \sin(2x),~ 0 \le x\le 2\pi$

Also find the respective maximum and minimum values.

Why not just use it as given?

Then $y' = 1 + 2\cos(2x)$ which has two zeros in $0\le x\le 2\pi$.
• July 12th 2011, 03:40 PM
amahindroo
Re: A Very Unique Maxima - Minima problem in Differential calculus
Quote:

Originally Posted by Plato
Why not just use it as given?

Then $y' = 1 + 2\cos(2x)$ which has two zeros in $0\le x\le 2\pi$.

Well, I had tried that also, but could get the 2 points as x = (pi)/3 and x = 2(pi)/3, which still is not as per the answer key given for that question.
• July 12th 2011, 04:07 PM