Maxima, minima, and the mvt application

In summary, maxima and minima are the highest and lowest points on a graph or in a data set, representing the extreme values of a function or data set. To find the maximum or minimum value of a function, the first or second derivative test can be used. The Mean Value Theorem (MVT) can also be used to find the maximum or minimum value of a function by determining the MVT point, which is where the slope of the tangent line is equal to the average rate of change of the function. The MVT application has many real-world uses, and it can be applied to all types of functions as long as they are continuous on a closed interval. However, there may be more than one solution using the M
  • #1
T Botha
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0
Hi there

I'm prepping for a big test tomorrow and I'm really struggling with this question:If f′′(x)≥−1, x belongs to (−15,15), and f′(1)=3, find the interval over which x is definitely increasing.I'm struggling with substitution because I just don't seem to have enough values. Is there a formula that gives an answer? Please let me know.

:( this is making me so sad.
 
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  • #2
Since [tex]f''(x)\ge -1[/tex], [tex]f'(x)\ge -x+ C[/tex]. Since f'(1)= 3, we know that C can be as large as 4. If [tex]f'(x)\ge -x+ 4[/tex] it will be positive for x less than 4.
 

1. What is the relationship between maxima and minima?

The terms maxima and minima refer to the highest and lowest points, respectively, on a graph or in a data set. In other words, they are the maximum and minimum values of a function or data set. The relationship between them is that they represent the extreme values of a function or data set, and can help us understand the behavior of the function or data set.

2. How do you find the maximum or minimum value of a function?

To find the maximum or minimum value of a function, you can use the first or second derivative test. The first derivative test involves finding the critical points of the function (where the derivative is equal to 0) and evaluating the sign of the derivative on either side of these points. The second derivative test involves evaluating the concavity of the function at the critical points. If the second derivative is positive, the critical point is a minimum, and if it is negative, the critical point is a maximum.

3. What is the Mean Value Theorem and how is it applied?

The Mean Value Theorem (MVT) states that for any continuous function on a closed interval, there exists at least one point within the interval where the slope of the tangent line is equal to the average rate of change of the function over that interval. This point is known as the MVT point and can be used to find the slope of the tangent line at that point. This theorem is often used in optimization problems to find the maximum or minimum value of a function.

4. How does the MVT application relate to real-world situations?

The MVT application has many real-world applications, such as determining the maximum or minimum speed of a moving object, the maximum or minimum production capacity of a factory, or the maximum or minimum cost of a product. Essentially, any situation that involves finding the optimal value of a function can utilize the MVT application to find the solution.

5. Can the MVT be applied to all types of functions?

Yes, the MVT can be applied to all types of functions, as long as they are continuous on a closed interval. This includes polynomial functions, trigonometric functions, exponential functions, and many others. However, it is important to note that the MVT may not always yield a unique solution, as there may be more than one point where the slope of the tangent line is equal to the average rate of change of the function.

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