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

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The discussion revolves around a request for feedback on completed optimization problems in a calculus class. A link to the work is provided for review. One participant notes that the overall setup and methodology appear correct but suggests maximizing \(d^2\) for simplification in the second problem. A mathematical expression is shared, indicating a solution involving square roots. The conversation emphasizes peer review and collaborative learning in calculus problem-solving.
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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.

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

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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