MHB Can You Prove This Using Fraction Division?

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The discussion focuses on proving a mathematical concept using fraction division, specifically involving the division of the numerator and denominator by the exponential function $e^{1/x}$. Participants are encouraged to provide a detailed step-by-step process to demonstrate the proof. The hint suggests that manipulating the fraction in this way may simplify the problem. The request for clarity indicates a need for thorough explanations in mathematical proofs. Overall, the thread seeks to clarify the application of fraction division in this context.
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can someone prove this and show the process in detail? Many thanks :)
 
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Hint: divide top and bottom of that fraction by $e^{1/x}$.
 

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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