Find the zeroes of the derivative?

  • Thread starter negatifzeo
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In summary, the problem is asking for the value of x that yields the global minimum for the function (3x^3-2x^2)-4x on the interval [-1,1]. To find this, we need to find where the derivative of the function is 0 and plug those values back into the original function. We also need to test both ends of the interval. The derivative of the function is 9x^2-4x-4, and the points at which it is 0 can be calculated exactly using the quadratic formula. This will give us the exact value of x for the global minimum. We can also use approximate values to determine if the minimum is at an endpoint or at one of the points
  • #1
negatifzeo
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Homework Statement


(3x^3-2x^2)-4x on the interval [-1,1]
Find the value of x that yields the global minimum.


Homework Equations


The derivative is 6x+4x-3


The Attempt at a Solution


Im not really sure. Don't I just find the zeroes of the derivative? It asks me for the EXACT answer which means the quadratic formula solution, but I am not sure how to.. Get there.
 
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  • #2


You can't always be sure that the minimum on [-1, 1] lies on a zero of the derivative. Find where the derivative is 0, plug those x values back into the equation, then test both ends.
 
  • #3


The max and min of a function on a give set can occur at three kinds of points:
1) At a point in the interior of the set where the derivative does not exist
2) At a point in the interior of the set where the derivative is 0
3) On the boundary of the set- which in the case of an interval consists of the two endpoints.

In this problem the derivative always exist so there are no points of type (1).

The derivative of (3x^3-2x^2)-4x is NOT " 6x+4x-3", it is 9x2- 4x- 4. The points at which that is 0 are indeed irrational but you can still calculate them exactly, using the quadratic formula. Since the problem asks only for the value of x at which the function has a minimum value, that should be enough. Even with approximate values for the function value at those points, you might be able to show that one of them gives the minimum value, or that the minimum is at an endpoint.
 

What is the purpose of finding the zeroes of the derivative?

The zeroes of the derivative are important because they represent the points where the original function has either a maximum or minimum value. This information is useful in understanding the behavior of the function and can be used to optimize processes in fields such as economics, engineering, and physics.

How can I find the zeroes of the derivative?

There are a few different methods for finding the zeroes of a derivative, including using the power rule, quotient rule, or product rule. Additionally, you can set the derivative equal to zero and solve for the variable, or use graphical methods such as the slope-intercept form of the derivative.

Why are the zeroes of the derivative important in calculus?

The zeroes of the derivative are important in calculus because they represent critical points on the original function, where the slope is either zero or undefined. These points can help us determine the concavity and direction of the original function, and can also help us find the maximum and minimum values of the function.

Can the zeroes of the derivative help me determine if a function is increasing or decreasing?

Yes, the zeroes of the derivative can help you determine if a function is increasing or decreasing. If the derivative is positive at a point, the function is increasing at that point. If the derivative is negative, the function is decreasing. The zeroes of the derivative represent the points where the function changes from increasing to decreasing or vice versa.

Are there any real-life applications for finding the zeroes of the derivative?

Yes, there are many real-life applications for finding the zeroes of the derivative. For example, in economics, finding the zeroes of the derivative can help determine the optimal production level for a company. In engineering, it can help optimize the design of a structure or system. In physics, it can help determine the position and velocity of an object at a given time.

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