How would you determine the maximum and/or minimum points of the graph y=xsin(1/x)?

Question

I've worked out y1= sin(1/x) - ((cos(1/x))/x)

and please I need to understand how you find the maximum or minimum, cause I have to write it in an assignment...

Please could you explain it in simpler terms, cause I'm only in high school and most of it goes completely over my head...

Answer 1

sin(1/x) (or cos(1/x)) is one of those crazy functions that oscillates all over the place as you get close to 0, which makes it really hard to work with. Fortunately it is bounded: its values are between -1 and 1.

What about xsin(1/x)? Is it bounded? Clearly, on [0,1) it is bounded between (-1,1) since |sin(1/x)|<=1 and x<1. What about for x>=1? Well, 1 is a bit less than pi/3, so the sin(1/x) component is constantly diminishing as x increases. But the x component is growing. What happens in the limit?

Limit as x->infinity of xsin(1/x) =
Limit as y->0 of sin(y)/y =
[Always remember, when a limit is sick, you take it to the hospital (take derivatives top and bottom)]
Limit as y->0 of cos(y)/1 = 1

The above shows that at x=1, the function is less than 1 and as x grows, the function goes to 1. But does it exceed 1 on this interval? If it did that, the derivative would have to be 0 on (1,infinity). You've already figured the derivative in your original problem statement so we set it equal to 0: xsin(1/x)=cos(1/x). Let's use z=1/x for z on (0,1). Then we are solving sin(z)=zcos(z) or z = tan(z).

z = 0 works. Anything else? Well, the derivative of z (with respect to z) is 1. The derivative of tan(z) is sec² z = 1/cos² z which is always greater than 1. Thus, there are no intersection points on (0,1). This means that on [1,infinity) zsin(1/z) has its minimum at x=1 and only appoaches its maximum (of 1) as x->infinity.

Now we consider what happens for x on [0,1). Since x<1 and sin(1/x)<=1 this means that xsin(1/x)<1 so we just need to find a minimum (which will be negative, of course). Since y=-x keeps getting closer to 0 as x->0, the first 0 derivative that we encounter as x goes from 1 to 0 will be give us the minimum.

We again look at where z = tan(z). It didn't happen on (0,1). However, as z increases to π/2, tan(z) goes to infinity and as z goes past π/2, tan(z) starts from -infinity, increasing to 0 as z gets to π, and then increasing to 1 as z gets to π+π/4, and then to √3 as z gets to π+π/3 > √3. So z=tan(z) will have an intersection point at a little greater than 4π/3. That is to say, the final 0 derivative of xsin(1/x) will be at x just a little less than 3/(4π) [just over .222], so that the minimum value of xsin(1/x) is approximately -3/(4π) [just over -.222]

In conclusion, the maximum of x*sin(1/x) is 1 as x-> +/- infinity and the minimum is achieved at |x| just greater than .222 with a value of about -.217

Answer 2

You get the minimum and the optimal of the graph whilst the derivate is null. f(x) = 3x² - 15x + 9 ? f'(x) = 6x - 15 f'(x) = 0 ? 6x - 15 = 0 ? 6x = 15 ? x = 15/6 ? x = 5/2 f(5/2) = 3(5/2)² - 15(5/2) + 9 f(5/2) = 3(25/4) - (seventy 5/2) + 9 f(5/2) = (seventy 5/4) - (one hundred fifty/4) + (36/4) f(5/2) = (seventy 5 - one hundred fifty + 36)/4 f(5/2) = - 39/4 ? that's the factor (5/2 ; - 39/4)

Answer 3

y = x sin(1/x)

For one thing, this is a tough graph to determine local maxima of.

y' = sin(1/x) + x cos(1/x) (-1/x^2)
y' = sin(1/x) - (x/x^2) cos(1/x)
y' = sin(1/x) - (1/x) cos(1/x)

Equating this to 0,

0 = sin(1/x) - (1/x) cos(1/x)

0 = [x sin(1/x) - cos(1/x)]/x

Our critical values are found when y' = 0 or when y' is undefined. Equating the numerator to 0,

x sin(1/x) - cos(1/x) = 0

x sin(1/x) = cos(1/x)
x = cos(1/x)/sin(1/x)
x = cot(1/x)

And, as you can see, this yields an equation where we cannot isolate x, and finding the critical values becomes a nightmare.

In essence, a difficult function to graph and find local minima/maxima for.

Answer 4

y=x sin(1/x)
y'= sin(1/x) -cos(1/x) *1/x
You have to find out the points where y'=0 put 1/x =z sin z-z cos z= 0 wich leads you to the equation tanz=z with z not 0 The first root is z=4.4934,the next7.7252.As for small z (x large) z=2.086*10^-6.
There is no rule nor periodicity

Answer 5

The minimum is x=+-0.23 (roughly)
Download MathCurve and see the graph
(it is easy and free)
http://www.softempire.com/mathcurve.html

Answer 6

If you do know so calculus then the easiest way to answer your question is to differentiate and then find any zeros. Then differentiate once more to find its concavity at those points.