These numbers... something isn't right
Given that x is real, the minimum value of 1 6 8 1 x 2 + 1 6 + ( 1 7 − 4 1 x ) 2 + 2 5 can be expressed in the form a b where b is square-free and a is a positive integer. Find a + b .
The answer is 371.
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3 solutions
Can you explain how you got the second step? And what do the solutions of y correspond to?
There's a much nicer Calculus free way, by taking the geometrical interpretation of the question. Hint: 3 7 0 = 1 7 2 + 9 2 .
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I'm going to have to think about that hint a bit. "I'll get back to you on that".
In regards to the other question, there's something I've been using since high school days, which is
If a + b = 0 , then a 2 − b 2 = 0 .
By repeated use of this step, I can [but not always] remove pesky radicals. In this case, doing this twice gets you the quadratic equation
( 6 7 2 4 y 2 − 1 9 4 3 2 3 6 ) x 2 + ( − 2 7 8 8 y 2 + 8 3 0 8 2 4 ) x + ( − y 4 + 6 6 0 y 2 − 8 8 8 0 4 ) = 0
which leads to that radical as a function of y . This is simply looking at the same curve as a function of y , and when x has a single value (when this radical disappears), then it's a tangent point, i.e., that one value of y between where x has two values and where x becomes complex. I use this technique a lot in many other contexts. It's a great time-saver.
Let me editorialize a bit and comment that the technique I've been using since "high school days" is a great way of converting functions into implicit equations, so that one can get a fuller picture of what's going on. Instead of bits and pieces of "functions", we have complete curves in 2D space, as defined by implicit equations. Of course, that can introduce more unintended roots, but that's where understanding what you are doing matters.
Calculus-free way.
Take the geometrical interpretation.
Hint: What is the distance when we travel from point
(
0
,
0
)
to point
(
a
,
b
)
to point
(
1
7
,
9
)
?
You'd need to find a suitable
a
and
b
.
Oh, all right, Calvin. Sometimes when something like this happens, I feel real stupid. We can re-express the equation in this way
( 4 1 x − 0 ) 2 + ( 0 + 4 ) 2 + ( 4 1 x − 1 7 ) 2 + ( 0 − 5 ) 2
which is nothing more than finding the minimum total distances from
( 0 , − 4 ) to ( 4 1 x , 0 ) and ( 4 1 x , 0 ) to ( 1 7 , 5 )
and so the obvious minimum distance is
( 5 + 4 ) 2 + 1 7 2 = 3 7 0
couldn't the points be (0,0) (41x,4) and (17,-1) and then applying triangles inequality 1 6 8 1 x 2 + 1 6 + ( 1 7 − 4 1 x ) 2 + 2 5 > = 1 7 2 + 1
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yes i did that and got 291...what's wrong with this? @Tan Wee Kean
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@Aritra Jana @Vibhav Agarwal (41x,4) cannot be on the same line segment as (17,-1) and (0,0). The y value of (17,-1) and (0,0) is not positive whereas the y value of (41x,4) is positive
This method was what I intended the solver to use. Unfortunately, it seems there are more than one way to solve it...
DIRECT APPROACH USING MINKOWSKI'S INEQUALITY
( 4 1 x ) 2 + 4 2 + ( 1 7 − 4 1 x ) 2 + 5 2 ≥ ( 1 7 − 4 1 x + 4 1 x ) 2 + ( 4 + 5 ) 2 = 3 7 0
Calculus-free way:
Solve for x the following
1 6 8 1 x 2 + 1 6 + ( 1 7 − 4 1 x ) 2 + 2 5 = y
to get
x = ( 8 2 ( y 2 − 2 8 9 ) ) − 1 ( 1 7 y 2 − 5 0 6 6 ± y y 4 − 6 6 0 y 2 + 1 0 7 3 0 0 )
and then solve for y the following
y 4 − 6 6 0 y 2 + 1 0 7 3 0 0 = 0
to get
y = ± 3 7 0 , ± 2 9 0