Long Roots Scare Me
Find the minimum value of the following expression:
x 4 − 2 9 x 2 − 1 2 x + 2 6 1 + x 4 − 3 5 x 2 − 2 0 x + 4 2 4
where x belongs to the set of all real numbers.
The answer is 8.5513.
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3 solutions
Yash, you are right in saying that you want the shortest distance from ( 1 5 , 6 ) to ( x 2 , x ) and then to ( 1 8 , 1 0 ) .
However, this shortest distance is not the straight line distance from ( 1 5 , 6 ) to ( 1 8 , 1 0 ) , because the graph of ( x 2 , x ) does not pass between these 2 points.
It's like saying "The shortest distance from Mumbai to New York and then from New York to Calcutta is just from Mumbai to Calcutta". We need to check that New York lies beween these 2 cities, otherwise it is clearly not true.
Instead, we now need to find the shortest distance from the point, to the parabola, and then back to the other point. The geometric interpretation, is that we want the smallest ellipse with foci of ( 1 5 , 6 ) , ( 1 8 , 1 0 ) , which touches the parabola defined by y 2 = x . We know further that this happens when the two (conics) are tangential.
I have updated the answer to 8.5513. Those who previously answered 8 or 9 (answer had to be an integer) were marked correct. (I currently do not think that there is a nice way to deal with the calculation, even with the above geometric interpretation.)
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This is one of those problems where even when the answer is wrong, it's nevertheless still interesting to see how it went wrong.
Thanks sir for your explanation. I would remember that in future and sorry for the wrong answer.
This function is nowhere less than 8 . 5 5 1 3 3 . . .
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But I think sir my solution is fine. What's your solution?How did you got that value?
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Brute force, I simply differentiated it and solved for the minimum numerically. The problem with your thinking is that while it works for x , y , it doesn't work in this case. It's interesting to go into why it failed this time, but right now I have to be somewhere else all day.
As much as "bashing" is condemned here in Brilliant, there's a lot good to be said for that approach.
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@Michael Mendrin – I would have a really nice approach if better numbers were used :/
I'll modify the numbers and write my own similar problem with an actual solution. @Michael Mendrin
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@Nathan Ramesh – https://brilliant.org/community-problem/a-minimum-value-sum/?group=QBJtOAykRTUk
This is a different idea, although i have a very different idea
@Michael Mendrin – How did you use the first derivative test to yield 8.551?
Yes it is nowhere less than 8 .55
by mistakley I pressed 5 and it showed correct? the answer is wrong
One could alternatively bash this problem out by using tedious optimization skills and then find the minimum value of the expression. To check, once you find that x is approximately equal to 4, by using the first derivative test, since the expression can be considered as a function of x, f(4) changes from negative to positive, thus indicating a minimum at roughly 8.55. #CalculusBash #Optimization
I simply used a TI-83+. I think we are allowed calculators. Though the method given below is probably what Brilliant is meant for.
This solution is incorrect. See the discussion below to understand why.
Factoring the given expression, we get:
( x 2 − 1 5 ) 2 + ( x − 6 ) 2 + ( x 2 − 1 8 ) 2 + ( x − 1 0 ) 2
Now, we have to think geometrically and visualize the above expression as the Distance Formula on the Cartesian Plane.
Then, we get the coordinates as ( x 2 , x ) , ( 1 5 , 6 ) , ( 1 8 , 1 0 )
Joining all the points and applying the Triangle Inequality in the triangle formed above, we get the minimum value as
( 1 8 − 1 5 ) 2 + ( 1 0 − 6 ) 2
which is equal to 5