Supremum

Find the supremum of this function.

Contents

Favorite Function

Here is one of my favorite functions. What is its maximum?

f(x)=tansinxsintanx

Let's plot it with ezplot, which is pronounced easy-plot.

f = @(x) tan(sin(x)) - sin(tan(x))
ezplot(f,[-pi,pi])
f = 

    @(x)tan(sin(x))-sin(tan(x))

The function is very flat at the origin. Its Taylor series begins with x7. It oscillates infinitely often near ±π/2. It is linear as it approaches zero again at ±π. And, most important for our purposes here, ezplot has picked the limit on the y-axes to be between 2.5 and 3.

syms x
F = sym(f)
disp('taylor = ')
pretty(taylor(F,x,'order',10))
ylim = get(gca,'ylim')
 
F =
 
tan(sin(x)) - sin(tan(x))
 
taylor = 
 
      9    7 
  29 x    x 
  ----- + -- 
   756    30

ylim =

  -2.867712755182179   2.867712755182179

Calculus

We learn in calculus that a maximum occurs at a zero of the derivative. But this function is not differentiable in the vicinity of π/2. The most interesting thing about an ezplot of the derivative is the title. Trying to find a zero of diff(F) is meaningless.

ezplot(diff(F),[-pi,pi])

Sample

We can sample the function near π/2 to get a numerical approximation to the value of the maximum. Is that good enough?

x = 3*pi/8 + pi/4*rand(1,1000000);
y = f(x);
format long
smax = max(y)
smax =

   2.557406355782225

Think

The computer has been a help, but we can do this without it.

sinx1

so

sintanx1

and

tansinxtan1

Consequently

f(x)1+tan1

Supremum

But I want to be a little more careful. As x approaches π/2, tanx blows up. So f(x) is actually not defined at π/2. For the domain of this function, one of the less than or equals changes to just a less than.

sinx<1

tansinx<tan1

f(x)<1+tan1

The precise answer to my original question is that this function does not have a maximum. It has a "least upper bound" or supremum, the smallest quantity that the function does not exceed. The sup is:

supf(x)=1+tan1

Now we can take a look at the numerical value.

sup = 1 + tan(1)
sup =

   2.557407724654902




Published with MATLAB® 7.14

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