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Spring 2018, problem 58

Let a and b be two real numbers and let M(a,b)=max. Find the values of a and b for which M(a,b) is minimal.

Comments

Hubert
4 years ago

Call this M=\text{max}\{u; v\},

now with (s,p)=(a+b,ab), AM-GM's \Rightarrow~4p\le s^2\Rightarrow~~

2M(a,b)\ge u+v=3(s^2-2p)+2s\ge \frac 32s^2+2s

=\frac 32(s+\frac 23)^2-\frac 23\ge -\frac 23=2M(-\frac 13; -\frac 13), the answer.

Any chance of expanding this answer with some explanatory notes ?

I'm probably not the only one who's not familiar with the terminology.

For example what does "AM-GM's" mean?

Thanks, Phil.

philboyd2 4 years ago
TME102475
4 years ago

The function M(a,b) is minimized when both of its arguments equal each other. That is:

3a^2 + 2b = 3b^2 + 2a \Rightarrow 3a^2 - 3b^2 + 2b - 2a = 0 \Rightarrow 3(a+b)(a-b) - 2(a-b) = 0 \Rightarrow (a-b)[3(a+b) - 2] = 0

or when: a = b (i), a+b = \frac{2}{3} (ii). Substituting (i) into either argument results in the quadratic equation:

f(a) = 3a^2 + 2a

whose first & second derivavtives yield:

f'(a) = 6a + 2 = 0 \Rightarrow a = -\frac{1}{3};

f''(-\frac{1}{3}) = 6 > 0 (hence, a global minimum)

which produces the critical pair P_{1}(a,b) = (-\frac{1}{3}, -\frac{1}{3}). Likewise, substituting (ii) produces:

g(a) = 3a^2 + 2(\frac{2}{3} - a) = 3a^2 - 2a + \frac{4}{3};

g'(a) = 6a - 2 = 0 \Rightarrow a = \frac{1}{3};

g''(\frac{1}{3}) = 6 > 0 (also a global minimum)

and the critical pair P_{2}(a,b) = (\frac{1}{3}, \frac{1}{3}). Checking both critical pairs against M(a,b) now results in:

M(-\frac{1}{3}, -\frac{1}{3}) = 3(-\frac{1}{3})^2 + 2(-\frac{1}{3}) = \boxed{-\frac{1}{3}}.

M(\frac{1}{3}, \frac{1}{3}) = 3(\frac{1}{3})^2 + 2(\frac{1}{3}) = \boxed{1}.

Since M_{P_{1}} \le M_{P_{2}}, \boxed{a = b = -\frac{1}{3}} are the critical values.

Now this is a solution I can understand !

However, the rather bold statement at the beginning surely needs some explaining.

i.e. "The function M(a,b) is minimized when both of its arguments equal each other"

Phil.

philboyd2 4 years ago

Apart from the obvious mistake (a wrong choice between the two extremal points) this solution is identical to mine; perhaps more detailed, but still without a justification for the equal values of the functions at the minimum point. As explained in my reply to Hubert, if a function is bigger than the other one near a minimum point for M(a, b), such a function should have a local minimum in such a point; however none of our functions can have local minima.

Claudio 4 years ago
Poojanpujara
3 years ago
  • let M(a,b)=max(α,β) where α=3a^2+2b and β=3b^2+2*a.
  • First solve for α>=β which yields a set S(say) {a>=b intersection a+b>=2/3 } union {a<=b intersection a+b<=2/3}.
  • Our domain is now decreased!!
  • I mean that every value M(α,β) will be    covered in this particular set S (i should say a part of domain) of a and b, because other set L(say)(rest part of domain)consisting of β>=α is just mirror image of set S if we plot a and b on x and y axis respectively.And off course The value M(α,β) will also be same for mirror images.[3a^2+2b for (a,b)€S and 3b^2+2 a for (a,b)€L because in above cases just the value of a and b are interchanged.] *** Note that now we just have to minimize α for set (domain) S.
  • Let m=3a^2+2b...now This is a parabola Whose equation is y=(m/2)-(3/2)x^2.
  • We have to minimize m (right??) so putting x=0 in above equation we get y=m/2.Which means that We have to decrease y coordinate(of a point where parabola cuts y axis) as much as possible.Plot the set S on coordinate axis with a and b on the x and y axis respectively and plot the parabola.But one should see that it should not be out of domain(I mean that choose minimal of y coordinate in such a way that the corresponding values of a and b should be in the domain S.)So after plotting This parabola and set S, we will come to know that the minimum value of m will occur when the line y=x will be tangent to the parabola y=(m/2)-(3/2)x^2 ...which will give m= -(1/3) which will give (3x+1)^2=0 ...giving x=y=-(1/3). It is solution that has never supposed anything !!(like other)Thanks for reading.You can ask by replying this answer if i was unable to satisfy you.
Claudio
4 years ago

The minimality of M(a,b) forces 3a^2+2b=3b^2+2a, say 3a^2-2a=3b^2-2b.

E.g. solving in a we get a=\frac{1\pm(3b-1)}3 say a=b or a=\frac23-b.

The first case bring to minimize 3b^2+2b and choosing b=-\frac13 we get the value -\frac13.

The second case asks for the minimum of 3b^2-2b+\frac43 and choosing b=\frac13 we get the value 1.

Thus \min M(a,b)=1, reached with a=b=\frac13.

Check that M(-\frac 13; -\frac 13)=-\frac 13, and how do you show that minimality yields equality of the terms ?

Hubert 4 years ago

modified on february 11

@Hubert: thank you for signalling my mistakes.

In particular I forget to say that functions must agree unless the minimum is reached in a local minimum of one of them.

My answer can be adjusted, but anyway your solution is better than mine

Claudio 4 years ago

Both a,b has same answer.. Btw thanks for solving it. Regards: Amaze Product

rdpforsale 5 months ago