MHB Solving for x: x^2 + (x-a)(a+1)/(x-a)

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The discussion revolves around the expression x^2 + (x - a)(a + 1)/(x - a), with participants questioning the intent behind it due to the absence of an equals sign. Clarification is sought on whether the objective is to simplify the expression or if additional context is needed. The lack of a clear goal leads to confusion regarding the next steps in the mathematical process. Participants emphasize the importance of providing complete information for effective problem-solving. Overall, the conversation highlights the need for clarity in mathematical discussions.
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x^{2} + (x - a) (a+1)/(x-a)
 
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prasadini said:
x^{2} + (x - a) (a+1)/(x-a)

Hi prasadini,

What is the goal here? There is no equals sign in the above, so if the goal to simplify? Is there something missing?
 

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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