Local Minima of y=x^3-2x^2-5x+2

In summary, to find the local minima or maxima of a function, you first find all the critical points by setting the derivative of the function equal to 0 and solving for x. Then, to determine whether a critical point is a local minima or maxima, you can use the first or second derivative test. For a polynomial function, the first derivative will also be a polynomial and therefore defined for all real numbers. The Hessian, or second derivative, can be used to confirm whether a critical point is a minimum or not.
  • #1
daniel69
10
0
for the equation... y = x^3 - 2x^2 -5x +2

is its local minima at (2.120,-8.061)

Thanks
 
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  • #2
How do you find local minima/maxima? First find all the critical points. HOw do you find critical points?
1.f'(x)=0
2.f'(x) does not exist
since your function is a polynomial it means that also it's derivative will be a polynomial of a less degrees, so it will be defined for all real numbers.

Now, after you find the cr. points, how do you distinguish whether it is a local minima or a local maxima?
SInce it is a cubic polynomial there will be max two local extremes.

Say c,d are such cr. points
then c is said to be a local minima if: let e>0, such that e-->0

so f'(c-e)<0,and f'(c+e)>0

and d i said to be a local max, if

f'(d-e)>0 and f'(d+e)<0.

Now do it in particular for your function.

Can you go from here?
 
  • #3
i just look at the hessian to figure out if it's a min or not
 
  • #4
Hi daniel69! :smile:

How did you get x = 2.120 ?
 
  • #5
JonF said:
i just look at the hessian to figure out if it's a min or not

Look at what?
 
  • #6
JonF said:
i just look at the hessian to figure out if it's a min or not
Since this is a function of a single variable, its "Hessian" is just its second derivative. However, that would be assuming that the x value given really does give either a maximum or a minimum- which, I think, was part of the question.
 

1. What is a local minimum?

A local minimum is a point on a graph where the function reaches its lowest value within a small interval of the point. It is lower than all the nearby points, but not necessarily the absolute lowest point on the entire graph.

2. How do I find the local minima of a function?

To find the local minima of a function, you need to take its derivative and set it equal to zero. Then, solve for the x-values that make the derivative equal to zero. These x-values are the coordinates of the local minima on the graph.

3. What is the equation for finding local minima?

The equation for finding local minima is f'(x) = 0, where f'(x) is the derivative of the function. Once you solve for the x-values that make the derivative equal to zero, you can plug them back into the original function to get the corresponding y-values.

4. Can a function have more than one local minimum?

Yes, a function can have multiple local minima. This can happen when the function has multiple "hills and valleys" or when the function is flat in certain areas.

5. How can I tell if a point is a local minimum on a graph?

To determine if a point is a local minimum on a graph, you can look at the slope of the function at that point. If the slope is positive on either side of the point, it is a local minimum. Additionally, you can also take the second derivative of the function at that point. If the second derivative is positive, the point is a local minimum.

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