Maximum distance of a line between two curves

Calculus Level 5

Consider two curves:

Curve A: $\ y = { x }^{ 2 }-8x-17$
Curve B: $\ y = 38x-4{ x }^{ 2 }-84.$

Take a point $L$ on curve A and a point $M$ on curve $B$ such that $LM$ is parallel to the $y$ -axis, and point $L$ is located below point $M$ .

What is the maximum length of $LM?$

The answer is 38.8.

1 solution

Varun Santhosh
May 11, 2018

$38x-4{ x }^{ 2 }-84-({ x }^{ 2 }-8x-17) = 46x-5{ x }^{ 2 }-67$

$\frac { d }{ dx } (46x-5{ x }^{ 2 }-67)=46-10x$

$46-10x=0$

$x=4.6$

$46(4.6)-5{ (4.6) }^{ 2 }-67=38.8$

The maximum length of LM is 38.8

That's actually the MINIMUM distance.

So, we can tell that the distance M of two points, as a function of x, is one polynomial minus the other: 5x²-46x+101. If we want to maximize that, we look at the points at which the derivative dM/dx = 0. In that case, 10x-46= 0, or x = 4.6. That's completely fine, but in order to find out if it is a maximum point, minimum point, or an inflection point, we need to take the second derivative. If it is greater than 0, that means the function is "accelerating" up, so that has to be a minimum point, since it is going up. If it is less than 0, ONLY THEN will it be a maximum point, since the function will be "de-accelerating". If it equals zero, it is neither a maximum or a minimum point.

Making this test, we conclude that d²M/dx² = 10 > 0, therefore it is a MINIMUM POINT.

Rafael Resener - 3 years ago

Hmm dunno why you look at the 2nd deravative. If we look at the first a sign check gives us that f' > 0 for x < 4.6 and f' < 0 for X >4.6, so it's a maximum. Next to that the resulting polynomial is concave, so it has a maximum. And the figure itself shows that the minimum distance is 0(namely the intersection points). There's just a mistaken in the answer or either in the question. As stated now the maximum value of LM is 38.8. But that's still not the whole truth, because the question never says we are bounded by the 2 intersections. If we are gonna be formal about it LM can be as long as you want it to be, but picking L and M left or right from the intersections.

Peter van der Linden - 3 years ago

How come $x^2-8x+17$ , when the problem states $x^2-8x-17$ ? That changes the answer from 38.8 to 4.8.

Michael Mendrin - 3 years ago

Your answer is wrong I am getting 38.1

Vijay Simha - 2 years, 12 months ago

Maximum distance of a line between two curves

The answer is 38.8.

1 solution

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