Minimum veritcal distance between two graphs.

[hobomoe]

Homework Statement

Calculate the minimum vertical distance between the cubic y=x^3-x^2-4x+4 and the parabola y=-2x^2+16x-30, when x is positive.

Homework Equations

y=x^3-x^2-4x+4
y=-2x^2+16x-30

The Attempt at a Solution

I have no idea at all how to work this out, so perhaps someone could walk me through it?

 

[hobomoe]

Made an attempt:
Since the x intercepts of each graph closest to each other are 2 and 3, the line must cut through 2.5. The coordinates for graphs are now (2.5,y1) and (2.5,y2). Substituting 2.5 into the equations gives the the y coordinates as 3.375 and -2.5. No need to use 2.5 to calculate distance since the gradient doesn't exist, so 3.374+2.5=5.875.

Not sure if this is right, but if it is could someone tell me how to work it out in a more calculus kind of way since this doesn't seem at all like calculus to me.

 

[Mentallic]

It's a nice attempt, but not exactly right. You can't make the assumption that since they're between 2 and 3 then it must be 2.5.
Take this graph, $$y=\frac{8x-16}{5-2x}$$. It's x-intercept is 2 but it has an asymptote at 2.5 so this is an extreme example of how it doesn't work.

If we have y=f(x) and y=g(x) then we let them equal each other so we have f(x)=g(x) then y=f(x)-g(x), we are finding x where the graphs of f(x) and g(x) are equal, or in other words, where f(x)-g(x)=0. But since in our case they don't cross, we are trying to find x where they are closest to each other. We are trying to find where y is a minimum.

 

[hobomoe]

How do you find the x intercept?

 

[Mentallic]

There is no x-intercept.
Again, in the function y=f(x)-g(x) where y is a minimum.

 

[hobomoe]

Whats g(x)? Is it f(x) for the 2nd equation?

 

[Mentallic]

Yep! Sorry I was assuming you knew about functions considering you're doing calculus already.

 

[hobomoe]

I do, but we don't use g(x), only f(x). Could you show me the first steps of working to get this? I'm pretty confused still.

 

[Mentallic]

Well, what's y=f(x)-g(x) ?

 

[hobomoe]

x^3+x^2-20x+34?

 

[Mentallic]

Right, so we have the graph y=x³+x²-20x+34. This graph tells us the distance between our function f(x) and g(x) since we are finding the difference between them. So at x=2, it will give us y=6 so that means the distance at x=2 between f(x) and g(x) is 6. How do you find where the minimum distance is?

 

[hobomoe]

I have no idea :(

 

[Mentallic]

Do you know calculus?

 

[hobomoe]

Yea but I'm having a mental block at the moment.

 

[Mentallic]

It's really simple! On the graph it tells us the distance between the two original curves f(x) and g(x). At x=2 we have y=6 which tells us the distance between the two curves is 6, at x=3 the distance between them is 10. Obviously the minimum distance between them is then the minimum point on the graph!
If I asked you to find the minimum of the parabola y=x²+2x using calculus, what would you do?

 

[hobomoe]

EDIT: Never mind with that, bit late

 

[Mentallic]

Don't think about finding what the actual distance is yet, we just want to find the x-value where the minimum occurs, which is where calculus comes into play. After we do this we'll substitute it back into equation to find the distance.

 

[hobomoe]

Is the minimum the turning point, so I need to find when the gradient equals zero?

 

[Mentallic]

You should be more confident in your abilities, rather:
"The minimum is the turning point, so I need to find when the gradient equals zero!"

:tongue:

 

[hobomoe]

f'(x)=3x^2+2x-20
I can't seem to manage factorizing this now, although I may be going in the wrong direction.

 

[Mentallic]

Maybe you can't factorize it? Use the quadratic formula.

 

[hobomoe]

X=2.27,-2.936

 

[hobomoe]

I use the 2.27 since the question asks when X is positive, then substitute X into f(x)-g(x) and that gives me 5.45 (2dp).

 

[Mentallic]

You should keep it in surd form. But we were told that x>0, so you can scrap the negative value. Now that you know where the minimum occurs, it's pretty obvious how close you are to the answer

Just as a note, cubics always cross the x-axis somewhere. Our difference function f(x)-g(x) crossed in the negative values of x, which means that f(x) equals to g(x) at some point (since their difference is 0) so if the question never restricted us to x>0 then the closest distance would be boring and just require us to solve the cubic f(x)-g(x)=0. This would be a harder question too...

 

[hobomoe]

I use the 2.27 since the question asks when X is positive, then substitute X into f(x)-g(x) and that gives me 5.45 (2dp).

Don't think you saw before you posted :P

 

[Mentallic]

Nope, it just took me longer than 3 minutes to write that down

That's it! You found the answer! Congrats.

 

[hobomoe]

Woo! Thank you so much for all your help! :D :D :D