Minimum and Maximum: (x^4+ x^2+5) /(x^4+ 2[x^2]+1)

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Firas

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Quadratic Equations – Minimum and Maximum


Find the minimum value of (x^4+ x^2+5) /(x^4+ 2[x^2]+1). [Ans : 19/20]


I know how to solve minimum and maximum for normal questions by putting them in the correct form but this is what i am kind of stuck on.
 
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You posted this under "arithmetic". There is no way to answer this using only "arithmetic"! Using Calculus methods, it is possible to show the maximum is 5, at x= 0, and there is no minimum- the function value approaches 1 as x goes to plus or minus infinity.
 
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Firas said:
Quadratic Equations – Minimum and Maximum


Find the minimum value of (x^4+ x^2+5) /(x^4+ 2[x^2]+1). [Ans : 19/20]


I know how to solve minimum and maximum for normal questions by putting them in the correct form but this is what i am kind of stuck on.
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What in the world do you mean by "normal questions"?
 
HallsofIvy said:
Using Calculus methods, it is possible to show … there is no minimum -- the f͏unction value approaches 1 as x goes to plus or minus infinity.
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I agree with that part, but the function does have a minimum value of 19/20. The graph's curve quickly drops below the line y=1, before turning around and approaching the line asymptotically from below. :cool:
 
Firas said:
Quadratic Equations …

(x^4 + x^2 + 5) / (x^4 + 2[x^2] + 1)
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Here are some quick notes about terminology. :cool:

That is not a quadratic equation.

It's not any kind of equation; it is an expression. (All equations contain an equal sign.)

We could call it a Rational function, in general, and we could call it a ratio of two quartic polynomials, in particular.

All quadratic equations can be put into standard form:

Ax^2 + Bx + C = 0 (where the coefficients A,B,C are parameters and A≠0).
 
mmm4444bot said:
Is this another practice exercise, for the Math Olympiad you mentioned earlier?
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yes.i know how to solve for the minimum value of quadratic equations in the form of a(b+h) + k. But, the degree of this question is 4.
 
Firas said:
yes.i know how to solve for the minimum value of quadratic equations in the form of a(b+h) + k. But, the degree of this question is 4.
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You know how to solve for the minimum value of quadratic equations in the form of a(b+h) + k? a(b+h) + k is not even a quadratic or even an equation so how can it be a quadratic equation?
 
Firas said:
yes.i know how to solve for the minimum value of quadratic equations in the form of a(b+h) + k. But, the degree of this question is 4.
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Do you pay any attention to the answers given?

There is a technique for finding minima and maxima of many types of functions. It is called differential calculus. You were told that in the very first answer in this thread. Take the first derivative of the function and determine at what values it is zero. Check that the sign of the first derivative changes on either side of each of those values to determine which are extrema and, if they are, of which type. Basic, basic stuff that you were told in the very first answer.

This function is not a polynomial function and does not have a degree in the sense that polynomials have a degree. It is a rational function. You were told that in the previous answer.

How is saying that you know how to solve for the extrema of quadratic functions responsive to being asked whether this is a practice problem for a math olympiad? Of what relevance is that knowledge when you have been explicitly told that the function is not a polynomial function at all?

Take the derivative and stop looking for formulas.
 
JeffM said:
Do you pay any attention to the answers given?

There is a technique for finding minima and maxima of many types of functions. It is called differential calculus. You were told that in the very first answer in this thread. Take the first derivative of the function and determine at what values it is zero. Check that the sign of the first derivative changes on either side of each of those values to determine which are extrema and, if they are, of which type. Basic, basic stuff that you were told in the very first answer.

This function is not a polynomial function and does not have a degree in the sense that polynomials have a degree. It is a rational function. You were told that in the previous answer.

How is saying that you know how to solve for the extrema of quadratic functions responsive to being asked whether this is a practice problem for a math olympiad? Of what relevance is that knowledge when you have been explicitly told that the function is not a polynomial function at all?

Take the derivative and stop looking for formulas.
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I wrongly wrote that form. I know what an equation is. When I answered I said, Yes ....., which means it was an olympiad problem. I haven't learned calculus so how can I understand that. If that is the only way, then I am ready to learn it. I was looking for an easier way of solving it. They don't expect students my age to know calculus.
 
Firas said:
I wrongly wrote that form. I know what an equation is. When I answered I said, Yes ....., which means it was an olympiad problem. I haven't learned calculus so how can I understand that. If that is the only way, then I am ready to learn it. I was looking for an easier way of solving it. They don't expect students my age to know calculus.
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Clearly they do!

I saw the following on the "About us" page associated with the sample problems you provided (I think that was you):

IJMO empowers students with deep conceptual understanding and logical thinking skills beyond Grade-level. This effectively stretches every student’s potential beyond Grade-level, allowing them to better apply much higher level logical and analytical skills to solve challenging Math Olympiad problems. Questions in IJMO are carefully designed to develop every student’s higher level of conceptual understanding and logical thinking skills.
​

It is actually not hard to teach yourself the basics of calculus (though this example is far from the most elementary); they apparently expect you to do so, because I can't imagine any other way to go about this unless you used a graphing calculator.

On the other hand, there is nothing like this among the sample problems; perhaps it is not truly representative. Where did you find it?
 
Dr.Peterson said:
Clearly they do!

I saw the following on the "About us" page associated with the sample problems you provided (I think that was you):
IJMO empowers students with deep conceptual understanding and logical thinking skills beyond Grade-level. This effectively stretches every student’s potential beyond Grade-level, allowing them to better apply much higher level logical and analytical skills to solve challenging Math Olympiad problems. Questions in IJMO are carefully designed to develop every student’s higher level of conceptual understanding and logical thinking skills.
​

It is actually not hard to teach yourself the basics of calculus (though this example is far from the most elementary); they apparently expect you to do so, because I can't imagine any other way to go about this unless you used a graphing calculator.

On the other hand, there is nothing like this among the sample problems; perhaps it is not truly representative. Where did you find it?
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I went for training classes. The teacher just guessed a few values and got the answer. That shows they don't actually expect us to know it, because they even said to us we don't need to. Ok, I'll try and teach myself calculus. Can you recommend some understandable sites?
 
Firas said:
I went for training classes. The teacher just guessed a few values and got the answer.
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I'm guessing that the training classes are to help you prepare for the Math Olympiad.

Guess-and-check is valid, if you can confirm your answer. In a few tries, I could understand finding an interval that contains the minimum, but to narrow down the answer to 19/20 without being able to use a calculator? I don't see it happening in just a few tries.


Firas said:
… they don't actually expect us to know it, because they even said to us we don't need to.
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Then why is the training class working on this exercise?
 
mmm4444bot said:
I'm guessing that the training classes are to help you prepare for the Math Olympiad.

Guess-and-check is valid, if you can confirm your answer. In a few tries, I could understand finding an interval that contains the minimum, but to narrow down the answer to 19/20 without being able to use a calculator? I don't see it happening in just a few tries.


Then why is the training class working on this exercise?
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I told you they probably expect us to guess a few values of x and then substitute them into the equation. When x = -3, the value is 19/20.
 
Firas said:
I told you they probably expect us to guess a few values of x and then substitute them into the e͏quation …
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You posted an expression. ;)

Yes, I understand how the guess-and-check process works, but not how the Math Olympiad works.

For example, if a rational function were to have a minimum at x = √(2) - 1/3, guess and check would not find it.

Has anyone told you that questions requiring methods not available to you are cooked up to give answers associated with Integers having small absolute value? In other words, you're being trained that it's fruitful to guess values like -3, -2, -1, 0, 1, 2, 3 during the exam as needed (to enable shortcuts) because the questions are designed that way?
 
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