When V'(x)=0: Exploring a Non Real Number

  • Thread starter PrudensOptimus
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In summary, the conversation discusses the function V(x) = x(10-2x)(16-2x) and how to find the values of x when V'(x) = 0. Through calculations and graphing, it is determined that the values of x are 2 and 6.6666666 (which is 20/3).
  • #1
PrudensOptimus
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V(x) = x(10-2x)(16-2x)

When i V'(x) = 0... x is always a non real number... I don't know why, can someone help me thanks.
 
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  • #2
Originally posted by PrudensOptimus
V(x) = x(10-2x)(16-2x)

When i V'(x) = 0... x is always a non real number... I don't know why, can someone help me thanks.

V(x) = (10x-2x^2)(16-2x)
V(x) = 160x - 20x^2 - 32x^2 + 4x^3
V(x) = 160x - 52x^2 + 4x^3
V'(x) = 160 - 104x + 12x^2
0 = 160 - 104x + 12x^2 I don't feel like doing quadratic equation so I just graphed it
x = 2, x = 6.6666666 (which is 20/3)

How did you go about getting your answers?
 
  • #3


When V'(x) = 0, it means that the derivative of the function V(x) is equal to zero. In other words, the slope of the function at that particular point is zero. This can happen at multiple points on a function, including at non-real numbers.

In this specific case, the function V(x) = x(10-2x)(16-2x) has multiple critical points where the derivative is equal to zero. These points are x = 0, x = 5, and x = 8. Therefore, when V'(x) = 0, x can take on any of these values, including non-real numbers.

To understand why this is the case, we need to look at the graph of the function. The graph of V(x) is a parabola that opens downwards, with its vertex at the point (5, 400). This means that the function has a maximum value of 400 at x = 5, and it decreases on either side of this point.

At x = 0 and x = 8, the function also has a slope of zero, but these points are not real numbers because they lie outside the domain of the function. This is because when x = 0, the function is undefined (you cannot divide by zero), and when x = 8, the function has a negative value, which is not possible for a volume.

Therefore, when V'(x) = 0, x can take on any of these values, including non-real numbers, because they are critical points of the function where the derivative is equal to zero. It is important to note that non-real numbers are still valid solutions in mathematics and can have real-world applications. I hope this helps clarify why x can be a non-real number in this situation.
 

1. What does it mean when V'(x)=0?

When V'(x)=0, it means that the derivative of the function V(x) is equal to zero at a specific value of x. This is known as a critical point or a turning point. At this point, the slope of the tangent line to the graph of V(x) is zero.

2. Why is a non-real number explored in this context?

A non-real number may be explored in this context as it could be a solution to the equation V'(x)=0. This may happen when the function V(x) is complex or has imaginary components. Exploring non-real numbers allows for a more comprehensive understanding of the behavior of the function.

3. What is the significance of V'(x)=0 in scientific research?

V'(x)=0 is important in scientific research as it helps identify critical points or turning points in a function. These points are key in understanding the behavior and properties of the function, which can provide valuable insights in various fields of study such as physics, chemistry, and engineering.

4. How is V'(x)=0 related to the concept of optimization?

V'(x)=0 is closely related to optimization as it helps identify the maximum or minimum points of a function. These points are important in optimization problems as they represent the most efficient or optimal solution to a given problem. By setting V'(x)=0 and solving for x, we can find these optimal points.

5. Are there any real-life applications of V'(x)=0?

Yes, there are many real-life applications of V'(x)=0. For example, in physics, V'(x)=0 can be used to find the equilibrium points of a system or the points where there is no net force acting. In economics, V'(x)=0 can be used to find the maximum or minimum points of a cost or profit function. These applications demonstrate the practical use of V'(x)=0 in various fields.

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