Absolute Maximum/Minimum Help

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In summary, to solve the equation (16x)/(x^2+4) for -5 (less than OR equal to) x (less than OR equal to) 5, use the quotient rule to find the derivative, set it equal to 0, and solve for x. The final equation should be a single fraction and should not result in complex solutions. It is important to be comfortable with both the quotient and product rules to avoid getting stuck in solving problems.
  • #1

Homework Statement

(16x)/(x^2+4) for -5 (less than OR equal to) x (less than OR equal to) 5

Homework Equations

Set the derivative of the equation equal to 0, solve for x to find the critical points, then plug and check for validity.

The Attempt at a Solution

I used product rule for (16x)*(x^2+4)^-1

I got the derivative as
(-32x^2)/((x^2+4)^2) + (16/(x^2+4))
which I then set equal to 0.

I then made an attempt to solve for x, but got x^2(48x^2+256)=-256, which I'm very unsure of, and am also not sure how to solve.
Help is appreciated.
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  • #2
You have made an error while solving your equation. Try to write your original derivative as a single fraction and I hope it would be more clear then what to solve.
(Your final equation gives complex solutions which shouldn't be the case ;) )
  • #3
I see now! I really need to get more comfortable with the quotient rule. I think I end up running myself in circles too often because I try to use the product rule, and then can't alter the problem to look like it would, if I had used the quotient rule.

Thank you very much!

1. What is an absolute maximum/minimum?

An absolute maximum is the largest value of a function over its entire domain, while an absolute minimum is the smallest value of a function over its entire domain.

2. How is an absolute maximum/minimum different from a local maximum/minimum?

A local maximum/minimum is the largest/smallest value of a function within a specific interval, while an absolute maximum/minimum is the largest/smallest value of a function over its entire domain.

3. How do you find the absolute maximum/minimum of a function?

To find the absolute maximum/minimum, you must first find all critical points of the function by taking its derivative and setting it equal to zero. Then, evaluate the function at each critical point and the endpoints of the domain. The largest/smallest value will be the absolute maximum/minimum.

4. Can a function have multiple absolute maximums/minimums?

No, a function can only have one absolute maximum and one absolute minimum. However, it may have multiple local maximums/minimums within its domain.

5. Why is finding absolute maximum/minimum important in mathematics and science?

Finding absolute maximum/minimum is important for optimizing functions in areas such as economics, engineering, and physics. It also allows us to understand the behavior and characteristics of a function, which can help in making predictions and solving real-world problems.

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