USCTrojanTennis' question at Yahoo Answers regarding optimization

In summary, the question is asking why the minimum and maximum points for y=f(x) also occur among the min and max points for y=f^2(x), assuming f(x)>0. The answer involves differentiating and using the chain rule and product rule to show that the extrema of f(x) and f^2(x) are the same. This concept can be further explored on a math help forum.
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MarkFL
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Hello USCTrojanTennis,

If we look at:

\(\displaystyle y(x)=f^2(x)\)

we find by differentiating with respect to $x$, using the chain rule:

\(\displaystyle y'(x)=2f(x)f'(x)\)

Since we are given \(\displaystyle f(x)\ne0\), we know that the roots of the derivative are simply those from $f'(x)$, thus we get the same critical numbers. To demonstrate that the nature of the extrema are the same in both cases, we may differentiate again, this time using the product rule:

\(\displaystyle y''(x)=2\left(f(x)f''(x)+f'^2(x) \right)\)

Now, at the critical points $x=c$, we of course have $f'(c)=0$, hence:

\(\displaystyle y''(c)=2f(c)f''(c)\)

Since we are given \(\displaystyle 0<f(x)\), then at the critical points the sign of the second derivative of $y$ is the same as the sign of the second derivative of $f$, and so we have shown that the function $f(x)$ and its square $f^2(x)$ have the same extrema.

To USCTrojanTennis and any other guests viewing this topic, I invite and encourage you to register and post other calculus problems in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.
 

Related to USCTrojanTennis' question at Yahoo Answers regarding optimization

1. What is optimization in the context of USCTrojanTennis' question at Yahoo Answers?

Optimization is the process of finding the best possible solution or outcome for a given problem or situation. In the context of USCTrojanTennis' question, it likely refers to finding the most efficient and effective way to improve the performance of a tennis player or team.

2. How is optimization used in the sport of tennis?

Optimization can be used in various ways in the sport of tennis. It can be applied to training methods, equipment choices, and even strategies and tactics during a match. By optimizing these factors, a player or team can enhance their performance and increase their chances of success.

3. What are some common techniques for optimizing performance in tennis?

Some common techniques for optimizing performance in tennis include analyzing and improving technique, developing a personalized training regimen, utilizing sports psychology strategies, and using data and analytics to identify areas for improvement.

4. How do you measure the effectiveness of optimization in tennis?

The effectiveness of optimization in tennis can be measured through various metrics, such as win-loss record, ranking improvements, and performance statistics. It can also be evaluated through the player or team's overall improvement and success in competitions.

5. Are there any potential drawbacks to optimization in tennis?

As with any strategy, there can be potential drawbacks to optimization in tennis. For example, focusing too much on optimizing certain aspects of the game may neglect other important areas. It's important to find a balance and not rely solely on optimization to achieve success.

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