Help with picking out which point has the most negative derivative

[Mr Davis 97]

Homework Statement

I am helping to tutor someone and we can't figure out the answer to the red part of this question. I thought that the answer would be x=3, but apparently, that's not right either. What do you guys think?

Relevant Equations

N/A

 

[fresh_42]

Which kind of function is it? What do you think or know?

 

[Mr Davis 97]

All we're given is that y = f(x). We don't know what kind of function it is. And it probably isn't $-\infty$ because that seems too complex for the nature of the question.

 

[fresh_42]

All we're given is that y = f(x). We don't know what kind of function it is. And it probably isn't $-\infty$ because that seems too complex for the nature of the question.

The function looks as a polynomial of degree $4$. I don't think that such a speculation is far fetched or complex.

 

[Mr Davis 97]

But I guess what I'm saying is that wouldn't this problem restrict itself to the interval $[0.5, 5]$ because that's all we're given?

 

[fresh_42]

But I guess what I'm saying is that wouldn't this problem restrict itself to the interval $[0.5, 5]$ because that's all we're given?

If we assume $x>0.5$ then the question is unanswerable due to a lack of information. It could be anywhre left of $1$ or around $3 \pm 0.5$. We need an assumption that it goes with $cx^4$ and $c>0$. Then $3cx^3$ will be the decisive term.

The image alone suggests $0.5$ or $3.5$.

 

[fresh_42]

But you can always draw tangents and measure the angle.

 

[Office_Shredder]

I would have guessed more like 3.1 or 3.2 than 3.

Just to point out, the picture you have is inputting 1.5, not 3 as your guess, in case you just messed up and tried the wrong think.

I agree this question is terrible.

I also agree 0.5 looks like a reasonable guess.

 

[MidgetDwarf]

The idea behind these type of problem is to remember that that the tangent to a point on a curve is equal to the derivative at that point. I am being a bit non mathy, so other more knowledgeable members please pardon me.
Anyways. With your initial choice of 1.5. If you draw a tangent line to the curve at the point 1.5, you get a horizontal line. Meaning the derivative is 0. Which is nonnegative.

As Fresh pointed, the values he gives gives what appears to give you the most "negative". In other words, what tangent to a point can you draw so that the angle is the steepest?This is what he means by measuring the angles ( I think, please correct me if I am wrong). Recall that the slope of a line is negative if it looks likes this \. The slope is positive if it looks like this / .

 

[MidgetDwarf]

Oh. Maybe this also works. I have not tested, but we have to make an assumption of how the curve looks like, based on the graph given.

Notice that the roots of the function occur at x= 1, 2,3,4.

Hence, f(x) = (x-1)(x-2)(x-3)(x-4). Expand the the right side of f(x). Then take the derivative of f. Now, input the values that Fresh stated. I am assuming, that it will collaborate what he stated ( I have not worked it out myself).

Generally, one does not proceed this way. This was only possible because the roots of curve were numbers we can easily read off the graph...

 

[Office_Shredder]

You know for a fact that the function is not (x-1)(x-2)(x-3)(x-4) because it's not symmetric about 2.5

 

[Mark44]

I would have guessed more like 3.1 or 3.2 than 3.

Same here. I think the goal of this problem is to get you to pick out by looking where the slope is the most negative. I don't think at all that the goal is to come up with a function, such as a polynomial, and take its derivative to fine where the slope is most negative.

OTOH, having you to pick out a number like 3.1 or 3.2 is not my idea of a well-designed problem.