# Really quick question about minimum of function

In summary, the conversation discusses how to find the minimum point on a curve given a specific equation. It is determined that the minimum point is at x=0 and the first derivative test can be used to prove this. Another question is then posed about finding the normal acceleration at x=0, but it is determined that there is not enough information to solve for this without knowing the position of the plane as a function of time.
!Really quick qestion about minimum of function

I ADDED ANOTHER QUESTION IN POST #3 ! Thanks for looking :)I am told the a jet's follows the curve $y=20(10^{-6})x^2+5000$ where y AND x are in feet. Basically all I need help with is for the first part of the problem, I need to find out where the jet is at its LOWEST point on the curve.

I am pretty sure this is at x=0. But how can I prove it? Usually I would just plug in zero for x. But this is not a function of time, it is of position, so I am not quite sure how to show that if I plug in a neg number I would not get a number less than 5000. I mean I know since it's an even function thath even numbers<0 for x will yield numbers>5000.

I was hoping to show this with Calculus rather than analysis, though maybe the latter is more reasonable in an engineering course?

Any thoughts are welcome, I can move on withthe problem regardless (since I could use the analysis) but I just want some input.

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In general find where dy/dx=0 and then check if that's a minimum. It doesn't matter that the equation is not a function of time. You are just finding the minimum point on a curve. In this case though you hardly even need to do that. You know x^2>0 unless x=0. So the lowest point is clearly at x=0, isn't it?

Dick said:
You know x^2>0 unless x=0. So the lowest point is clearly at x=0, isn't it?

Like I said, I know that it is at x=0, but I'm an idiotm so I would to prove that it is (to myself). So I see your point. 1st Derivative test does work.

ANOTHER QUESTION:

How about this now: I need to find the normal acceleration $a_n$ of a point traveling along said curve AT X=0.

Now I know that $$a_n=\frac{v^2}{\rho}$$

My teacher gave the hint $ds=rd\theta$. How can I find velocity from this? I would nee to write theta as a function of x?

Any ideas?

......

andrevdh said:
From a dynamical viewpoint one can say that since the jet is in level flight (derivative is zero) at the lowest point it follows that the jet is exerting no vertical force (the engine is horizontal). This means that only gravity is dictating its vertical acceleration.

? Gravity has absolutely no effect on the acceleration of the plane. It's not in free fall. It's following a fixed trajectory. You can't find the normal acceleration from the information you are given. To use v^2/R you would need to find the radius of curvature, R, of the curve at x=0 and then find v. But you have no way of finding v without knowing something about the position of the plane as a function of t. For all you know, it could just be sitting at x=0 without accelerating at all.

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## 1. How do I find the minimum of a function?

To find the minimum of a function, you need to take the derivative of the function and set it equal to 0. Then, solve for the variable. The value of the variable when the derivative is 0 will be the minimum of the function.

## 2. Can a function have more than one minimum?

Yes, a function can have multiple minima. These are called local minima and occur when the derivative of the function is 0 at more than one point. However, there can only be one absolute minimum, which is the lowest point on the entire function.

## 3. What is the difference between a minimum and a minimum value?

A minimum refers to the point on a function where the value is the lowest. A minimum value refers to the actual numerical value of the function at that point. For example, a function may have a minimum value of 2 at the point (3,2).

## 4. How do I know if a function has a minimum?

A function will have a minimum if its derivative is 0 at a certain point. If the derivative is never equal to 0, then the function does not have a minimum. However, it is important to note that the function may still have a minimum value, even if there is no minimum point.

## 5. Can a function have a maximum and a minimum?

Yes, a function can have both a maximum and a minimum. These are called local extrema and occur when the derivative of the function is 0 at multiple points. However, there can only be one absolute maximum and one absolute minimum on a function.

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