anonymous
  • anonymous
what is the smallest slope on the curve y=x^3-4x+1?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Also, at what point does the curve have said slope?
tkhunny
  • tkhunny
Define "smallest". Is 0 SMALLER than -3, for example?
anonymous
  • anonymous
I think it means "smallest" as lowest in its context.

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tkhunny
  • tkhunny
That doesn't mean anything. The choices are: Closest to zero. Greatest possible negative.
anonymous
  • anonymous
Greatest possible negative.
tkhunny
  • tkhunny
Okay, how shall we proceed? Do we get to use the calculus?
anonymous
  • anonymous
Yep, I'm in Calc 1, so that's what I'd prefer.
tkhunny
  • tkhunny
Fine. This is an interesting question because we don't much care about the original function. We are much more interested in the derivative. Please find the 1st Derivative and see if that helps answer the question.
anonymous
  • anonymous
I got Y prime=3x^2-4 for the derivative.
tkhunny
  • tkhunny
That's correct. No what? Reading the problem statement, we want to minimize THIS function. We want to know about the SLOPE of the derivative - not the value of the derivative. We're going to need the 2nd derivative. Pleese provide.
anonymous
  • anonymous
F double prime= 6x. But i'm guessing that won't do me any good either, so the third derivative is 6. Is my math right?
tkhunny
  • tkhunny
Why won't that do you any good? You want to know where a function has it's minimum. Isn't that the job of a derivative? If I just handed you this problem, could you solve it? Find the minimum value of f(x) = 3x^2 - 4 What would you do?
anonymous
  • anonymous
Plug in 0 for x?
anonymous
  • anonymous
Or find the derivative?
tkhunny
  • tkhunny
Very good. The ONLY point of confusion on this problem, and hardly anyone gets it on the first try, is that we are trying to minimize a function that we already know in our heads is a 1st derivative. If we can get over that, we'll be fine. Original Problem Statement: What is the smallest slope on the curve y=x^3-4x+1? Rephrase (specific): What is the minimum value of y'? Rephrase (Generic): What is the minimum value of f(x) = 3x^2 - 4?
anonymous
  • anonymous
So the minimum value of y' is 6x?
tkhunny
  • tkhunny
6x is not a value. I'm trying to get you to forget that it is y'. The minimum value of 3x^2 - 4 is -4. If you would like to find the derivative of f(x) = 3x^2 - 4 to determine that x = 0 is where the minimum value exists, then feel free to do that.
anonymous
  • anonymous
So to find the minimum value of f(x)= 3x^2-4, all I have to do is plug in a zero into that equation?
tkhunny
  • tkhunny
If you happen to know that x = 0 is the right place, yes. If you took enough analytic geometry before calculus, as it se3ems you have, this should be relatively obvious.
anonymous
  • anonymous
I'm sorry, I'm just not understanding why zero is the right place?
tkhunny
  • tkhunny
So, maybe you haven't had enough analytic geometry? Faced with the parabola, y = 3x^2 - 4, where is the vertex? Or, like you did up above, find where the 1st derivative is zero. Either way, you get x = 0 as the right spot.
anonymous
  • anonymous
Oh okay I understand now.

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