[Calculus] Finished optimization problems. Would someone please check them?

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    Calculus Optimization
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SUMMARY

The discussion focuses on two optimization problems completed for a calculus class. The user requests feedback on their methodology and solutions. A key hint provided for the second problem is to maximize \(d^2\) to simplify calculations by avoiding square roots. The user presents a specific equation, \(-\dfrac{1}{2x}x+3=x^2-1\), leading to the solution \(x=\pm\sqrt{\frac{7}{2}}\).

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  • Understanding of calculus optimization techniques
  • Familiarity with derivatives and their applications
  • Knowledge of quadratic equations and their properties
  • Ability to manipulate algebraic expressions
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  • Study optimization techniques in calculus
  • Learn about maximizing functions using derivatives
  • Explore the implications of using \(d^2\) in optimization problems
  • Review quadratic equation solutions and their graphical interpretations
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Students studying calculus, particularly those focusing on optimization problems, as well as educators looking for examples of student work in this area.

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I haven't checked every step, but the setup and overall methodology looks fine to me. Hint on Number 2: maximize $d^2$. Then you don't have to bother with square roots.
 
2.

$$-\dfrac{1}{2x}x+3=x^2-1\Rightarrow x=\pm\sqrt\frac72$$
 

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